AN ANALYSIS or HEAT AND MASS TRANSFER IN ATMOSPHERIC FREEZE-DRYING Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY GARY ARLYN HOHNER 1970 -~r.“‘ f f'. This is to certify that the thesis entitled AN ANALYSIS OF HEAT AND MASS TRANSFER IN ATMOSPHERIC FREEZE-DRYING presented ‘bg Gary Arlyn Hohner has been accepted towards fulfillment of the requirements for $41—— degree in M . 9. I4 CHAIM Major professor Date March 31, 1970 0-169 I' am‘mue BY v , HBAB & snus' I‘ BOOK mom mc. " PIER-ARV BINDERS' ISA.‘ IM‘W" .w 113' 11 _ ; I r 1" ‘C. WF‘I-g‘l'nt r F“I\ ‘- 'm&tut ton em"... :.- "I \ k . 1‘ ‘ SM 3. ac rial; .. I: _ C‘" 1.3 t;m:'.7q.t.,v.-, I ‘ . ». -\ ". ‘d h pan“: 7'5 “v“ w ‘ - ~' ~~' :"H ”“1 proce 1: v' ,- _... .' ' . ~ - ~-.:» I. Savor!” ;-:wcr , ”I: ,. ~.. “.4 It than“! , - H." . ”7.“. wugscmr “ .til‘hflted 11....1 YP-I. '_‘_ . 3‘. U. “,‘r ,M‘, I“ ‘n. . . x .. ' 3—2"- .3 m “fictive f rh . .: Mun.“- “a. emu:- I" I V l W“ prunes: ; ~. -._.~.VJ;I;‘;, -.; fray a; 11!. And sow!“ a a Mitxyh' maize. ' r); ant-wan 4;” inlnna‘ figs: ms} '93,! (fiwlf'l'u (Irving-W at ’0 muted wlmu ‘9' the Lug-we fillet-ate m. W ”Chanda r-f Md' aid *5 ataxia: .m ism for: ' ~~ 'QILausl'gatcl ABSTRACT AN ANAIXSIS OF HEAT AND MASS TRANSFER IN ATMOSPHERIC FREEZE-DRYING BY Gary Arlyn Hohner Atmospheric freeze-drying is the process of dehydration by sublimation conducted at atmospheric preSSure. The conventional sublimation dehydration process used in food and other biological products is conducted at very low pressures, usually below the ,- triple point of water (4.58 mm Hg). Widespread application of the . conventional process to convenience foods is limited by economic fictors. Several investigators have noted that if sublimation could :be conducted at atmospheric pressure expensive vacuum equipment could be eliminated and the process could be made continuous. The objective of the research conducted was to analyze the 2 u rate- limiting process in atmospheric freeze-drying of precooked ‘7 beef by deriving and solving a mathematical model of the process . Vagdyevaluating the internal heat and mass transfer coefficients of the product. The computed values of the transport coefficients fill . J._r r r Gary Arlyn Hohner .the product. The mechanism of mass transport was assumed to be yater vapor diffusion through the air-filled, porous layer. Heat transfer through the porous layer was assumed to be by conduction. Complexity of the model precluded a closed form integration, so the techniques of numerical analysis were used to obtain a solution. Three tranSport parameters of the numerical solution were evaluated using the statistical technique of nonlinear estimation. These parameters were the effective thermal conductivity, the structural constant of the internal mass transfer coefficient,, and the surface mass transfer coefficient. The effect of air temperature, system pressure and orientation of the fiber struc- ture on each estimated parameter was analyzed statistically from results of experimental tests. Only the effect of fiber orienta- tion on the structural constant was significant at the 90% con- fidence level. The effective conductivity was found to have a mean value of .0001ca1/cm-sec-OC. The structural constant had a mean value of .81 for transport parallel to the fiber structure and .62 perpendicular to the fibers. Analysis of the practical operating space of the process variables for atmospheric freeze-drying was accomplished by trans- forming the proven, one-dimensional, numerical solution into an approximate three-dimensional solution. The rate of atmospheric freeze-drying in cubical samples of precooked beef was found to ' be directly and strongly dependent on air temperature. When the dimensionless ratio of surface to internal mass transfer coef- 'r;§ieients exceeded 100, the rate of drying was confirmed to be trsely related to the square of the sample thickness. Gary Arlyn Hohner ; Ited to small particle size and high air flow rates. ;h£nCd-bsd or spray dryers are possible equipment configura- Approved: Bfl. W 21.4,. Major Professor 3/70 Department chairman v; ,.t .* lx'nvu‘. in port‘s! Mums“. AI ANAEYSIS OF HEAT AND MASS TRANSFER IN ATMOSPHERIC FREEZE-DRYING By Gary Arlyn Hohner A THESIS Submitted to .‘ ' 'é ltichigan State University ' * . - in partial fulfillment of the requirements for the degree of DOCTOR 0P IHIIOBOIII ‘1970 . , ‘q‘V‘I'n' ‘ he outbox- ‘ r-: . . - ;. .' , Aleochtl F. 1 ' , r t. ..' '. .a ..»-_ a' 1 l.‘ Decency n .. . , . - hu- , - .‘.,‘ ,. IWAWC'IF .~ ' 4 1‘ sf ‘nf- ' ‘ r . . In . ' ' qut‘ ‘ M100 {‘1 :lr' ' 1 ,' I‘y‘. . -MO l.’ ALF") “v. . . I . av H-ZV-H ~ ". .:"'t--’; ,‘1‘« 0.00:, fiSI'LC‘I" . . '2 .4 a: 4‘ It, '3. t, “mun, z‘x- . of Food 6. I..- 3 .v w» 1.- " M. utrda‘u‘- meant-rel. A special (If-TC -,. .. apnm .. u} s a To: Gay, Gail and Douglas ‘1 ”043.7. Prairsr. i :sx. net 1‘? 1:43. 9' r v-mml-‘try .4 '1. en. '_ V. -ll‘ ”0 Miner-ms v :w '41.“ u- ’Iv ‘3"4" 2- mew-r111 3 > -- h “ wihflmllihl ‘ ,_ 1;. ~37-1L; :1 'mv' »_ 1,}: j.»,.'g.,n a“ “£1.13 . 5% g , ~~10 50 so hoax : l'r ;~.;-u.;_ "I: ‘mmietton is otF." arms-acted to 7w. F u “1;. Mg...“ Ad (.\3A- .\... ‘ ‘-‘- of. Agricultural Engineering, and Dr. H, ‘ icnhew, ' . 1‘3"? m fiafeuoc, Food Scienteg for ,8» . -..; oar neiuam ._ ,'; N. " " re 2' Q he. centre: the rev-Iarch 1:?me as wrr two-rm. hr . “a?“ of coupes-at inn am: can am.- I. worsens}: “"3. 515" ‘9', ”one can be conductor" v ‘ a author wishes so a“... V». $78!.le14, . q .,. h the put severe! ”as. ._‘- ACKNOWLEDGMENTS The author wishes to express appreciation to Dr. D. R. Heldman, Associate Professor, Agricultural Engineering and Food Science Departments, and Chairman of the guidance committee, for his continuous guidance and assistance throughout planning and execution of the graduate program represented by this thesis. Gratitude is also extended to Dr. F. W. Bakker-Arkema, Associate Professor, Agricultural Engineering, and Dr. W. M. Urbain, Pro- fessor of Pood Science, for serving on the guidance committee. A special expression of appreciation is due Dr. J. V. Beck, Associate Professor, Mechanical Engineering, for exemplary teaching and numerous consultations in the areas of numerical solution of mathematical models and parameter estimation on which this thesis is so heavily based. Appreciation is also expressed to Dr. C. W. Hall, Chairman and Professor, Agricultural Engineering, and Dr. B. S. Schweigert, Chairman and Professor, Food Science, for resources and facilities with which to conduct the research program and more important for fostering the type of cooperation where meaningful interdisci- '.7 . - plinary graduate programs can be conducted. Finally, the author wishes to thank Dr. H. B. Pfost, Pro- ”Circum- n. s. Brooker, Mr. M. R. Gould, Mr. E. c. Rupp and Dr. :'§§;~L. Clark who, over the past several years, have expressed en- ’1~;§:~Iagement and support for completion of this academic endeavor. -’ ‘ 111 TABLE OF CONTENTS Page RACKNOWIEDGMENTS ... ....... ............... ......... ..... iii IIST OF TABLES ........................................ Vi LIST OF FIGURES ....................................... vii LIST OF APPENDICES .................................... ix NOMENCLATURE .......................................... x Chapter I. INTRODUCTION ................................ 1 11. REVIEW OF LITERATURE tease-annoooaaeaeoaaeeea 6 AtmOSpheric Freeze-Drying ................. 6 Mechanisms and Parameters of Heat and Mass Transfer in Freeze-Dried Foods ..... 9 Freeze-Drying Rate ........................ 13 111‘ THEORY ........OIOIIIOOIOIIOIOIIOIOOIIOOOIOI. 17 The Mathematical Model .................... 17 The Mass Transfer Coefficient ............. 24 The Finite-Difference Model ............... 27 Sensitivity Analysis and Estimation of Model Parameters ........................ 33 IV. EXPERIMENTAL DESIGN AND PROCEDURES .......... 39 Experimental Design: Sensitivity Analysis ................................ Preparation and Assignment of Samples ..... Selection of Test Conditions .............. Experimental Apparatus and Procedures: Equilibrium Studies ..................... 50 Experimental Apparatus and Procedures: Rate Studies ............................ 54 5&8 :;‘v‘; mmMmL SowTION ...-......CUDIUIUOCIOI 60 iv mummDISwSSION ....IIOOOOOCOOOOOOOOO. Simlation of Atmospheric Freeze-Drying in filtee Dimensions IICOIIIOIOIOIOOOOOIOOII. . J mavmlyaia of Atmospheric Freeze-Drying in A" .m' Qt. gmcwsross, MI 2 "‘.... i The Parameter Estimates ... ‘CCur‘cy of the “Odel IO‘OODIOIOIIOOCOOOOOOO‘ VI mic‘l smples I...’........I..II3......- Page 73 73 83 87 99 101 LIST OF TABLES Numerical Values of Physical Constants Used in the Mathematical Model ................ Summary of Parameter Estimates and Variance °f|theResidua18 ...OIOOOOOOOOOIIOO0...... Sumary of Analysis of Variance in the Paramter Estimates ......IIOODOCCOIIOUOI. Page 64 76 77 Figure 5.3 LIST OF FIGURES Schematic Diagram of Atmospheric Freeze- Drying l.It.lIIOIIOIIOOOOIOCUOOOIIIIO0.... Geometric Basis for the Numerical Solution .... Dimensionless Sensitivity Coefficients of Unknown Parameters for Freeze-Drying at .97 AtmosPhere Pressure .................. Dimensionless Sensitivity Coefficients of Unknown Parameters for Freeze-Drying at .58 AtmOSphere PreSSure .................. Contours of the Risk-Function Surface in the Vicinity of the Computed Minimum ......... Schematic Drawing of Experimental Apparatus for Equilibrium Moisture Content Studies . Schematic Drawing of Experimental Apparatus for Atmospheric Freeze-Drying Rate Studies Sample Holders, Spring and Thermocouple Assembly for Drying Rate Studies ......... Equilibrium Moisture Isotherms of Freeze-Dried Beef Belowzero 0C ......ICCIIOIIOCIJOI... Computed Profiles of Vapor Pressure and Temperature for Typical Conditions of Atmospheric Freeze-Drying ................ Comparison of Predicted Drying Curves of the Proposed Model and the Paeudo Steady-State Model in One-Dimensional Samples ......... Comparison of the Proposed Model to Experimental Results at .97 and .58 Atmosphere Pressure IIIII......IIOOOCIOOOOUIIIII....O Comparison of Solutions of the Three-Dimensional Model to Experimental ReSults in Cubical samples a...ease-ssoaseeeeeeaso-nas-aeasae vii Page 18 28 42 43 46 51 55 57 62 69 71 85 89 Page Predicted Effect of Air Temperature on Atmospheric Freeze-Drying Rates in . One-Centimeter Cubes of Precooked Beef ... 92 Predicted Effect of System Pressure on ' Atmospheric Freeze-Drying Rate in One- Centimeter Cubes of Precooked Beef ....... 93 Predicted Effect of Sample Size on Atmospheric Freeze-Drying Rate in Cubes of Precoohd Beef .....IIOOOOOOIOOOOOIOII. 95 6.6 Dimensionless Time Elapsed at M - .1 in Cubes of Precooked Beef as a Function of H Under Atmospheric Freeze-Drying ... 96 6.7 Predicted Maximum-Rate Drying Curve for Atmospheric Freeze-Drying of Cubical smples of hecooked Beef ......OCOCOUOOI' 98 ‘rixg,i viii IJST 0F APPENDICES Page Derivation of the Energy Equation, Equation (3.1) OI..IOOOOOOOIOOIOIOIOOIOI 105 Derivation of the Mass Transfer Equation, Equat1°n(3.2) 'IDOC-......IOIIOIOOOOOOO 107 Derivation of Finite-Difference Approx ima- tions of the Heat and Mass Transfer Equations , Equations (3 .18) and (3 .19) . 108 Computer Program MAIN and Subroutine MODEL . 110 Summary of Results from Parameter Estima- tion Tests IOID........IIOOICOI-OIOIIOII 116 NOMENCLATURE Area, cm? Tridiagonal matrix, equation (3.21). Tridiagonal matrix, equation (3.21). Column matrix, equation (3.21). Constant, equation (2.1), cm2. Constant, equation (2.1), cm. Constant, equation (2.2), dimensionless. Specific heat of dry product, constant pressure, calorie/gm-°C. Specificoheat of water, constant pressure, calorie] 8M' Co Mutual free-gas diffusivity, air and water vapor, cm lsec. DP, cmz-atm/sec. Effective water vapor transfer coefficient in the porous zone, defined by equation (3.15), gm/sec- cm-atm. Porosity of the porous zone, dimensionless. Position of ice-vapor interface, dimensionless. Function of - Defined by equation (3.23). Defined by equation (6.2). Surface heat transfer coefficient, calorie/cmZ-sec-oc. Surface mass transfer coefficient, 3m/cm2-sec-atm. -._Identity matrix~ Index. Index. Knudsen diffusivity for air, equation (3.10). Defined by equation (3.10). Knudsen diffusivity for water vapor, equation (3.10). Effectiveothermal conductivity, porous zone, cal/cm- sec- C. Effective thermal conductivity, frozen core, cal/cm- sec-°C. Nuuber of experimental points in a test. Moisture content, dry basis. Dimensionless mean moisture content. Molecular weight, air, gm/gm-mole. Molecular weight, water, gm/gm-mole. Adsorbed moisture content in equilibrium with satu- rated water vapor, dry basis. Initial moisture content, dry basis. Moisture content at the ith node and nth time frame in.the numerical solution. Total number of space nodes in the numerical solution. Space node nearest the ice-vapor interface in the numerical solution. Mass flux rate, gm/cmz-sec. Matrix, defined by equation (3.32). Index. Order of - Total pressure, atmosphere. Partial pressure of air, atmosphere. Partial pressure of water vapor at the ith node and ' nth time frame in the numerical solution, atmosphere. xi Saturated vapor pressure, atmosphere.‘ 'Partial pressure of water vapor, atmosphere. Partial pressure of water vapor in the air stream, atm. I Column matrix of water vapor pressures in the nth time frame. Heat flux rate, calorie/cmZ-sec. Universal gas constant, cm3-atm/mole-OC. Risk function of - Relative humidity, dimensionless. Column matrix of sensitivity coefficients. Dimensionless sensitivity coefficient of parameter (1), defined in Figures 4.1 and 4.2. Half-thickness of the sample, cm. Temperature in the porous zone, 0C. Temperature of the air stream, 0C. Temperature of the frozen core, oC. Temperature at the ith node and nth time frame in the numerical solution, 0C. Time, sec. Distance, cm. .Column matrix of experimental observations. 'De(Ae>/D)RT ax sz u T x P P where: Kw = cll/fiWM—w , Ka = cl/ffflfi; and Km = P—" Ka + P—“ Kw. Three experimentally determined constants, C0, C1 and C2 which are functions of the porous structure are necessary to completely describe the flow of gases through a porous medium under all flow conditions. The above authors presented convincing experimental proof of the validity of the derivation and its superiority over similiar expressions based on familiar capillary tube assumptions. In atmOSpheric freezeadrying the transported gas is water vapor, and the stagnant gas is air, or some inert gas. The maximum possible total pressure differential across the porous zone is approximately 4.5 mm Hg compared to a total pressure of over 700 mm Hg. Thus, the second term of equation (3.10) was considered negligible. The remaining term was written xi - -02 M“ d—P‘L (3.11) C2D K dx ’ (— + J“— RT Kw 1% Km Pa 1: K3 PV K3 K3 w — B — J— = - — _ —- — = —— but K P +-P K 1 P (1 K ) and K M (3.12) W W W W a -c D dP so a.= 2 “k: —". (3.13) C23 ( w dx E_— + P-Pv 1 - fi_) RT w a For the gas mixture of air and water vapor (3.13) gave ‘02” M dP a. = w —" . (3.14) - dx czD [E-— + P - .209 Pl] RT K v . W 26 Sandall, g; 31. (1967) evaluated Kw in the outer breast meat of turkey to be approximately 20 cmZ/sec for tranSport parallel to the fibers and approximately 10 cm2/sec perpendicular to the fibers. These data were obtained with nitrogen as the inert gas. Assuming the same order of magnitude for K.w in beef with air as the inert gas, equation (3.14) was further simplified. By defini- tion C2 is less than one and 5 is approximately .2 cmz-atm/sec for air and water vapor at zero 0C. By order of magnitude analysis on the effective mass transfer coefficient, the following evalua- tion was obtained: = C2D Mw e (.8)(.2) _ .209 4.5 ° E 10 + 1 760 3R1 Clearly, the first and last terms of the denominator are not only D (3.15) insignificant compared to P, but also of opposite sign. As a result, the effective mass transfer coefficient for atmospheric freeze-drying was written D = ._ ___ (3.16) It is unlikely significant thermal gradients exist in food products sized for practical application of atmospheric freeze- drying; if they do equation (3.16) can be modified to account for thermal gradients by the familiar 7/4 power rule (Eckert and Drake, 1959). c D' / G Mr”4 D =—2—T—)7“. 2 (317) e PRT T' .774 ° ' PR1 27 The Finite-Difference Model With larger, faster and more flexible digital computers which have been developed in recent years have come more sophis- ticated and reliable numerical methods. The present level of computer capability and numerical solution technique has elevated applied numerical solutions from crude apprOximations to an accurate and preferred method of solving real—world problems. Numerical solution of partial differential equations was discussed in an excellent manner by Smith (1965). TWO general approaches exist for transforming a complex physical situation and its associated mathematical model (partial differential equations in this case) into algebraic, finite-dif- ference equations which constitute the numerical model. The first approach can be described as merely writing finite-difference equa- tions to correctly approximate the partial differential equations without regard for the physical problem. The other approach is to derive the numerical model directly from the physical process by application of conservation laws to a differential volume increment. The current situation is an excellent example of the usefulness of the latter approach. The domain of the equations which were to be solved to describe the process of atmospheric freeze-drying was the porous zone of the sample. At time equal to zero the domain was also zero. Furthermore, the domain continuously increased until the ice front reached the centerline. Finite-difference approximations of the partial differential equations were written at (m)-space grid points or nodes across the domain of the partial differential 28 equations. Obviously, since the domain increased, either the nodes had to shift position or additional nodes had to be added to the domain as time progressed. The later case was chosen. Geomet- rically the numerical solution was represented as shown in Figure 3.2. F—Ax—i . // l l 0 WT— V— f i o l: l mf l mf+1 I q 1 ] / floating grid point flxed grid p01nt Figure 3.2. Geometric Basis for the Numerical Solution Fixed position grid points were equally spaced from center- line to surface of the Sample. An additional floating grid point remained on the ice-vapor interface (f). As the interface passed a fixed grid point that node was included into the numerical solution. No volume of the porous zone was associated with the floating grid point; it represented conditions of the ice core. The point nearest the interface represented a variable volume which increased as the interface receded. When the interface passed a new grid point the variable volume of the previous nearest grid point was truncated to normal size, and the dependent variables at the new grid point assumed a linearly interpolated value between the value of the floating grid point and node in the newly truncated grid volume. Computation continued until the floating grid point reached the 29 centerline at which time all grid points represented a constant volume. Turning attention to numerical approximation of the partial differential equations, the domains of the independent variables, dimensionless time and distance, were divided into small increments A9 and A¢ respectively. Using the procedure presented by Crank and Nicolson (1947), the mass transfer equation was approximated by finite-difference equations. eM n 9 + + + +1 ( w + dM) (Pn 1 _ P2) = %(Pn 1+ n 1 2Pn +Pn n n RT Psat dr 1 i+1 i-l- i 1+1+P1-1'2Pi): (3-18) 1 i = m m, 15’ “HP ' ' ' n = 1,2,3 . . . . Equations (3.18) were written from a conservation of mass in the differential volume element, Ado, rather than being written directly from equation (3.4). Ideally the energy equation should also be approximated by the Crank-Nicolson method since this method has the smallest truncation error of commonly used finite difference approximations (Smith, 1965). However, in spite of the fact that the Von Neumann stability analysis (Smith, 1965) predicted the Crank-Nicolson approximation of equation (3.1) to be stable, trial solutions demonstrated the numerical solution to be unstable. The apparent inconsistency is probably explained by the nonconstant coefficients and expanding domain of the model which were not considered in the Von Neumann analysis. For these reasons the energy equation was transformed to finite-difference equations using the backwards- difference technique (Smith, 1965). 30 n+1 n _ n+1 n+1 n+1 (cpdmicpwwri 4'1) ' ZZ (T1+1+T1-1 2T1 ) + Cpgz Pn+l_Pn+l n+1 Tn+1 + n+1 n 3 19 4 (1+1 i-1)(Ti+1' 1-1) Auvpmi "’41): (. ) i = mf, mf+1, . . . m, n = 1,2,3, . . Boundary conditions at the Surface and the ice interface were transformed to finite-difference equations in a similiar manner. Details of the numerical derivations are shown in Appendix III. Equations (3.18) and (3.19) each represented (m-mf+1) algebraic equations which were solved simultaneously for the temperature and vapor pressure at each node point in the porous zone during time frame A6. In each time step a solution was obtained for equations (3.18) first. Rearranging (3.18) the following equations were obtained: n+1 eMfi p dM n n+1 n+1 'ZPi-1+2(RT+P dr+z Pi 'ZP1+1= sat 3.20 zpn +251E"-+—2——d—lvl-znP“+zpIn ( ) i-l RT p dr i i+l’ sat i i = m£,mf+1, . . . m. Equations (3.20) were represented in matrix notation as A-Pn+1=B°Pn=C (321) — ..v _ w _, . where A and g are both tridiagonal matrices. A11 quantities on the right-hand side of equation (3.21) were known allowing them to be evaluated to yield a known column matrix, Q. Row operations were performed on tridiagonal matrix A to eliminate the lower 31 diagonal which in turn allowed direct evaluation of the vapor pres- Pn+1 sure matrix , —v , by back substitution. The appropriate matrix row and back substitution operations are summarized in an algorithm presented by Smith (1965). The moisture content at each Space node in the (n+1)th time frame was immediately computed from the equilibrium moisture relation- ship as soon as the vapor preSSure was known. The temperature column matrix in the (n+1)th time frame was computed in the same manner as outlined above for vapor preSSure. Vapor pressure and adsorbed moisture content at each space node for the (n)th and (n+1)th time frames were substituted into equations (3.19). The equations were rearranged into tridiagonal matrix form and solved for the temperature column matrix by use of the above mentioned algorithm. Use of finite-difference equations for solution of partial differential equations raises questions of accuracy and stability. Stability implies convergence of the numerical solution to the actual solution. Absence of stability usually reSults in increas— ing oscillation of the numerical solution about the true solution. As previously mentioned not all methods of finite-difference approx- imation are equally stable or accurate. The Crank-Nicolson approx- imation used for the mass transfer equation has the smallest truncation error of any commonly used numerical approximation. This error is 0(A¢)2 + 0(Ae)2 for approximation of second order, diffusion-type, partial differential equations in one dimension. As previously mentioned this approximation is stable for all values of A¢ and A6 with constant coefficients and a constant domain; 32 however, accuracy decreases with increasing increment size. The backward difference approximation provided a stable representation of the energy equation; however, it has no minimum truncation error (Smith, 1965). From these facts it is obvious improved accuracy can be obtained from the numerical solution, within the limit of computer round-off error, by reducing the increment sizes. The cost of improved aCCuracy is computation time. In the final analysis no presently available analytical stability investigation is highly definitive in complex models Such as the one under con- sideration. The investigator is left with the practical method of trial solution of the model while varying increment size until an acceptable compromise between solution time and accuracy is obtained. Selection of appropriate increment sizes is discussed in Chapter V. Numerical approximation of convective boundary conditions such as equations (3.5) are not stable for all values of A9, the time increment. Also, the truncation error of the finite-difference equation for the convective boundary condition is of the order of A¢ which is, of course, larger than O(A¢)2. Stability of the entire numerical solution depended upon stable numerical approx- imation of the convective boundary condition. This approximation is derived below and the maximum allowable dimensionless time step, A6, was computed for a given dimensionless space increment. Con- sidering the numerical approximation of the mass transfer Surface boundary conditions: 33 eM n .75?n+1 + .25 Pn+1 - .75 Pn - .25 P” _A_Q( w + 1M.) ( m m-l m m- 1) = 2 RT fisat dr m A9 (3.22) D Pn+1 _ Pn+1 Pn _ Pn h s _e(m-1 m +£24)-.2. Pn+1+Pn_2P 2D A¢ A¢ 2D m m va Rearranging (3.22) into tridiagonal matrix form gave eM n eM n Pn+1[:.25(-—" + J— 9!) - z] + P2“ [.75(-—‘1 + i— i!) + z + H] = m m-l RT Psat dr RT Psat dr m n th dM n n eM dH.“ Pm-1 .25(-R—f '4' LP a) + Z + Pm [.75(—£RT + LP a?) - Z - H] sat m sat m + 2PaH’ (3.23) Deae hDs A9 where: Z ='--- and H = -5_K$_ . D(A¢) Smith (1965) has shown equation (3.23) is stable provided the co- efficient of B: is always positive. Then, m 75(fM"-+-JL—-9!)n ' RT P dr A9 s n sat m (3.24) De h s m + ....D_ D(A¢)2 DA¢ gives the relationship between Ad and as which satisfies stability requirements. Sensitivity Analysis and Estimation g£_Mode1 Parameters A primary objective of this research was to use experimental atmospheric freeze-drying data and the mathematical model to obtain estimates of tranSport parameters for atmospheric freeze-drying in beef. The discussion of this section will be limited to the 34 mathematical basis of parameter estimation in nonlinear models and the associated subject of sensitivity analysis. The goal of sensitivity analysis is to predict what values of the independent variables correSpond to conditions where the model is most sensitive to changes in the value of the parameters and, therefore, at what point the most useful data can be taken to evaluate a particular parameter. The subject of parameter estimation in nonlinear models is relatively new. Few references are dated prior to 196C. Draper and Smith (1966) gave a one-chapter introductory coverage of the subject and provided a sizeable list of references. They pointed out three general methods are used for obtaining parameter estimates from nonlinear models. First, the method of linearization, which is presented later in this section, has been used by Beck (1966) and by Pfahl and Mitchell (1969) in parameter estimation from numerical solutions of models of partial differential equations in the field of heat transfer. Second is the method of steepest descent, which may have some advantages over the linearization (Gauss-Newton) method when the initial parameter estimates are considerably different from the final optimum estimates. The third (Marquardt's compromise) combines desirable features of both pre- vious methods by using the method of steepest descent during early cycles of parameter improvement and gradually switching to the method of linearization. Marquardt's compromise has been written into a general computer routine, called GAUSHAUS, for parameter estimation in nonlinear models (Meeter, 1964). Use of this program is dis- cussed in Chapter VI. 35 For the situation under consideration experimental data correSponding to the model solution were dimensionless mean moisture content as a function of dimensionless time. Solution to the model can be written in terms of the same variables, 1 - we. ) M(e’§) = f +j. -M'—-.B- d®g (3.25) f o where g_ is the parameter column matrix. The observed data were represented as Yj =fij(g) + ej , j = 1,2,3....L, (3.26) where was the random error of measurement associated with h the (j)th experimental reading. The optimum values of the para- meter matrix were determined by minimization of the deviation between the experimental data and the model according to some pre- selected criterion. For the preposed model the selected criterion was to minimize the sum of squared deviations. Based on this selection a risk function was defined as, Rm>=q-Em»W4q-Emn. (aw) where Y was the covariance matrix of the observations. It can be demonstrated that weighting the quadratic risk function with the inverse of the covariance matrix produces a minimum variance estimate of the parameter matrix (Deutsch, 1965). From a practical point of view it is probably impossible to know the value of all elements of the covariance matrix. If they were known it would be a large computational task to obtain the inverse since the matrix has L2 elements; where L is the number of experimental 36 measurements. This difficulty is commonly circumvented by assuming the experimental data are independent observations. The assumption of independence implies zero covariance between two separate observa- tions and reduces the covariance matrix to diagonal form. The matrix can then be easily inverted. The assumption of independence between any two experimental readings was made in the analysis pre- sented herein. Following previous derivations (Draper and Smith, 1966; Beck, 1969A), the minimum variance parameter vector was derived. The optimum value of the parameter vector exists when the gradient of the risk function is zero, that is, at the minimum of the risk function. _. -1 _ = 2 ' - = . V‘BR(B') LEV M (m)! (X HQ.» 0 (3 23) By expanding the expression for the model in Taylor series and retaining only the first two terms, the model was written as 11(5) = Ego) + §(§0)(s - so) (3.29) where: Sac) = 18E@‘fio. The expressions represented by §(fio) are called sensitivity co— efficients. Their magnitude indicated how sensitive the model was with respect to a given parameter at a particular value of the independent variable, dimensionless time. Substitution of equation (3.29) into (3.28) and rearranging gave Econ”); - §'<30>i"[fi] = o. (3.30) Solving for g, the improved estimate of the parameter matrix, gave 37 a - so = [§'(no>i'1_s_mo)] 'ls'mom‘lm - also». (3.31) Repeated calculation by reinserting §_ into (3.31) as Bo led to a further refined parameter matrix. This procedure was repeated until some minimum change in any element of the parameter matrix was not exceeded. Alternately, computation could have been ter- minated if some preselected change in the total sum of squared deviations between the data and the model was not exceeded. Whether or not the calculations suggested by equation (3.31) yield accurate estimates of all elements of the parameter matrix simultaneously, or whether the equation even exists depends upon the sensitivity coefficients. Inspection of equation (3.31) shows the answer to both questions is found by analysis of the matrix )1 = s_'(20>i'1_s.<30>. (3.32) If the determinant of N_ is zero the matrix is singular and the right side of equation (3.31) does not exist. The magnitude of the determinant of N_ is dependent upon the values of the sensi- tivity coefficients. If any combination of the sensitivity co- efficients of the parameters are linearly dependent two or more columns of N. are identical and §.13 singular. Further dis- cussion of this point is presented in the next chapter. Values of the sensitivity coefficients as functions of the independent variable, dimensionless time, were computed from the numerical solution of the mathematical model. 5 a 514(913fi) a M(Gj,81(1+6)38fi) ’ M(ej :3.) 11 as, be, (3.33) 38 where: 5 was a small number such as .01, i = 1,2, or 3; the number of parameters in the model and 6 JAG. Use of the sensitivity equations as computed by equation (3.33) in design of experiments is discussed in the next chapter. CHAPTER IV EXPERIMENTAL DESIGN AND PROCEDURES Experimental design may be considered as two subjects when applied to problems of estimating physical parameters in mathe- matical models of biological processes. One type of experimental design makes use of the sensitivity coefficients derived in the previous chapter. The sensitivity coefficients can be used to define an optimum experiment for producing data to estimate the model parameters. An experiment can be optimum in the sense that data obtained from it will yield least variable estimates of the model parameters, provided the mathematical model is correct. The sensitivity coefficients may also be used to determine if all of the parameters can be simultaneously estimated, if nonuniform weighting of the data is desirable or necessary, and how accurately the parameters can be estimated. Experimental design also refers to the concept of random selection and assignment of experimental units to the various test conditions. Random selection of test samples is especially important in studies such as the current research. Almost invariably the complexity of biological systems exceeds the ability of the in- vestigator to explain all of the possible variables in mathematical terms. Therefore, relatively minor variables are omitted from the model or are assumed to be constant. Random selection of samples from a relatively large population of similiar samples allows 39 40 statistical analysis of the results to minimize the effects of external and uncontrolled variables. Experimental Desigg; Sensitivity Analysis The mathematical model presented in Chapter III contains two internal tranSport parameters to be estimated from experimental data which are functions of the structure and composition of the product. These parameters are effective thermal conductivity of the porous zone, k, and the structural constant, C in the effective 2, mass transfer coefficient. The most accurate data for estimating a particular parameter are obtained under experimental conditions which maximize the sensitivity of the dependent variable to the parameter of interest. In freeze-drying heat and water vapor must be transferred through both internal and external transfer resistances. Since internal transport parameters were of primary interest, the optimum experi- ment was one which caused the dependent variable, dimensionless mean moisture content, to be dependent on internal tranSport resistances. This was accomplished by minimizing external re- sistances to heat and water vapor transfer. Therefore, the optimum atmospheric freeze-drying experiment for estimating internal trans- fer parameters was conducted with maximum possible air flow over the sample surface. The upper limit on air flow rate in the current research was dictated by experimental equipment limitations. If sufficient air velocity over the sample could have been used to assure the surface transfer coefficients to be effectively infinite relative to the internal transfer coefficients, the exter- nal parameters could have been neglected, and, in turn, the surface 41 boundary conditions for the mathematical model would have been simplified. Calculations based on the Reynold's analogy for a flat plate indicated a dimensionless ratio of external to internal mass transfer coefficients of approximately ten for the air velocity used. This value was not sufficiently large to warrant neglecting the surface mass transfer parameter, hD; thus, it was included with the internal parameters to be evaluated from the data. The surface heat transfer coefficient, h, was not required since surface temperature was monitored experimentally and used in the numerical solution as a boundary condition. By assuming the proposed mathematical model accurately simulated the process in question, the numerical solution of the model was used to compute sensitivity coefficients as a function of elapsed time for each of the three unknown parameters. Figure 4.1A shows the absolute value of dimensionless sensitivity coef- ficients for each parameter for typical test conditions at .97 atmosphere total pressure (the approximate atmOSpheric pressure at East Lansing, Michigan). Similiar results for a total pressure of .58 atmosphere are shown in Figure 4.1B. Sensitivity coef- ficients in Figures 4.1A and 4.13 were computed for a sample half- thickness of .45 cm and air temperature of -2.8°C. Values of the unknown parameters were estimated at .012 gm/cmz-sec-atm for hD, .8 for c and .00015 cal/sec-cm-OC for k. 2 The magnitude of the dimensionless sensitivity coefficient * for C2’ SC , increased monotonically with elapsed time until the 2 ice-vapor interface reached the centerline. At this time free ice no longer existed in the sample, and the boundary condition at the 42 Dimensionless Sensitivity Coefficients m N \O m Q I L l I A O‘ L .od .ousmmuum mumnamoao< um. um wcwmunuonomum pow muoooamumm csocxcs mo muamwoflwwoou mufl>wuwmcmm mmoacowmcoEHa ~83. mmmacofimcmflwa u o .fiuHmCom mmoHconcoEwn .ma.q muswwm page mamasoHocmeQ . o .oaa .oma .ooH .ow .ow .os .ON .y ’ - D l. , if P r h D .- D - on X oi A 3| l 'K (D X I OH «m nuauuoo aannsron ssaIuoisuamIa N- 44 centerline abruptly changed from a saturated temperature and vapor pressure condition to an adiabatic condition. Change in the boundary condition caused an immediate drop in the magnitude of all sen- sitivity coefficients. They all decreased to zero as the sample reached equilibrium. The magnitude of SED increased rapidly to a peak early in the drying process at both pressure levels. It then declined slowly to a nearly constant value while most of the free ice was removed from the sample. Absolute value of SzD was approximately one order of magnitude less than 822 over most of the drying time. The sensitivity coefficient for effective thermal con- * ductivity, S , also increased until all free moisture had been k * removed; however, the magnitude of SR was approximately one- * fifth that of SC throughout most of the process. This dif- 2 ference in magnitude indicated freeze-drying is mass transfer controlled at both pressure levels considered. Decreasing the system pressure increased the effective mass diffusivity while the effective thermal conductivity remained relatively constant. This caused the process to shift in the direction of heat transfer . . . * control with a corresponding increase in S and a decrease in k 822. These results can be noted in Figures 4.1A and 4.18 by comparing the coefficient magnitudes between the two figures at the same dimensionless time. Further reduction in system pressure to the range of conventional vacuum freeze-drying would have caused the process to be almost totally heat transfer controlled. The pro- cess would then have been relatively more sensitive to the heat transfer parameter, k, and less sensitive to C2. The magnitude 45 k illustrates that no single experiment can be designed to simul- * * of S would then have been greater than 8C2. This discussion taneously maximize the sensitivity coefficients of both heat and mass transfer parameters. Thus an experiment cannot be the optimum experiment for estimating both heat and mass transfer parameters at the same time. Another potential difficulty in simultaneously obtaining accurate estimates of k and C is the proportional relationship 2 x * that seems to exist between SR and SC in both Figure 4.1A and 2 4.18. Perfect linear dependence between the two sensitivity co- efficients would cause N_ to be singular and prevent simultaneous estimation of any combination of parameters which included both k and C2° To further access the seriousness of the near linear * * k and SC the risk function of the experimental 2 data obtained in test number 4 (See Appendix V) was plotted over dependence of S the domain of k and C2 in the vicinity of the minimum computed by the nonlinear estimation procedure. The surface mass transfer coefficient was held constant at the estimated value, .0116 gm/cmz- sec-atm. Contour lines of the risk function surface in the vicinity of the computed minimum are shown in Figure 4.2. These results indicated a single minimum does exist and confirmed that s: and 322 are not perfectly linearly dependent. Nonlinear estimation of all three parameters was possible but the distended nature of the contours of the risk function surface indicated limited accuracy in the computed estimates of k. Magnitude of the sensitivity coefficients indicated how accurately the corresponding parameter could be estimated for a 46 1.75 q 7-99 3.0 -70 06 9.09 3.94 .02 .45 1,93 1055- ——h1 kxlO4 - cal/cm-sec-OC 1.15 . -121211 6-50 1.35 . jump 4.91 bank} 1.04 9 ,_L.38 24\\ 7 .95 . w 9.10 4.5\99 1.56 I I I I I .65 .75 .85 .95 1.05 62 Figure 4.2. Contours of the Risk-Function Surface in the Vicinity of the Computed Minimum. 47 given level of accuracy in determining the dependent variable. The dependent variable, dimensionless mean moisture content, was determined from sample weight which was read from a scale with minimum divisions of .05 mm. Associated with this reading was a Spring constant of .0243 gm/mm. Therefore, sample weight was determined to approximately .001 gm. Initial sample weight was approximately one gram. Accuracy of the parameter estimates was approximated from the average magnitude of the reSpective sen- sitivity coefficients shown in Figure 4.1A. Rearrangement of the dimensionless sensitivity coefficients gave the minimum detectable error in the parameter for a given average magnitude of the sen- sitivity coefficient. '- AC G L“ - .2, then —-3 = '001 = .005 AM _ Ak _ .001 = '— K Ah 0M _ 0 = .001 _ These results indicated all parameters could be estimated to within a maximum of three percent of their true value. Finally, the sensitivity coefficients of Figures 4.1A and 4.1B could be used to weight the experinental data so that data taken when a particular sensitivity coefficient was maximum‘were given more weight in determining the parameter values than data taken when the sensitivity coefficient was smaller. Due to the relatively uniform values of the coefficients for the parameters of the proposed model it was decided that nonuniform weighting of 48 the data was not necessary. Preparation and Assignment 2: Samples Experimental samples were prepared from the loin eye muscle of beef, longissimus dorsi. A section of the muscle, grade U.S.D.A. Choice, was obtained from MSU Food Stores and roasted at 163°C until the temperature at the center of mass reached 740C. Average composition of 10 samples of the cooked beef was 9.8% fat (ether extract) with a moisture content of 150% d.b. The cooked meat was then frozen at -290C and later cut into approximately one-centi- meter cubes with an electric band saw. Special effort was made when cutting the cubes to align the natural fibers of the meat with the planes of the cube. The cubes were wrapped in foil and stored at -29OC in a sealed container until needed for a test. Cubes in which the fibers of the meat projected at an oblique angle to the faces of the cube were discarded. From the remaining population cubes were assigned randomly to a particular set of test conditions. Selection gthest Conditions Conditions under which experimental data were to be obtained were selected to investigate the practical operating space of atmo- spheric freeze-drying in precooked beef and to adequately test the pr0posed mathematical model. As cited in Chapter II, previous investigators have found atmospheric freeze-drying rates are greatly accelerated by small increases in air temperature over the range from -200C to zero oC (Woodward, 1961; Lewin and Maletes, 1962). Preliminary tests conducted in the current research indicated significant product shrinkage in beef dried with air temperatures 49 above zero oC. Calculation of the amount of unfrozen water in beef from apparent Specific heat data indicated a large change of unfrozen water per unit change in product temperature in the range from -100C to zero oC (Hohner and Heldman, 1970). Presumably the mechanism of moisture transfer during dehydration and qualitative factors in the dehydrated product may be significantly altered by the amount of unforzen water present in the product during drying. These considerations and the fact that sublimation is exceedingly slow at temperatures below -100C led to selection of two levels of air temperature, -2.80C and -8.20C. Woodward (1961) has also shown that practical applications of atmospheric freeze-drying in foods are limited to small sample sizes. A11 drying results reported herein were obtained in samples approximately one-centimeter thick. By definition, atmOSpheric freeze-drying is conducted at or near one atmosphere pressure. Nevertheless, for purposes of adequately testing the prOposed mathematical model, data were obtained at two levels of total pressure, .97 atmosphere and .58 atmosphere. Data by Harper (1962) indicate the value of effective thermal conductivity in freeze-dried beef is the same for both of these pressure levels, approximately 1.5x10-4 cal/cm-sec-OC. From the discussion of Chapter III related to simplification of the effective mass transfer expression, it may be noted that the model predicts vapor diffusion is still the predominate mechanism of tranSport at .58 atmOSphere; however, the effective mass transfer coefficient is inversely related to system pressure. 50 Estimation of C2 from data obtained at both pressure levels represented a strenuous test of the validity of the model. The model assumed this parameter was a function of the structure of the product only and not of system pressure; therefore, C2 should be estimated as the same value at both pressure levels. The same is true for thermal conductivity. Finally, because of the prominent natural fiber structure of beef it was assumed that heat and mass transfer rates might be different for tranSport perpendicular to, as Opposed to parallel to, the fiber structure. Therefore, both orientations were in- vestigated. Three repetitions of each of six combinations of the above variables were conducted. Experimental Apparatus and Procedures: Equilibrium Studies As discussed previously in Chapters II and III, some fraction of the total moisture in beef is adsorbed on the non- aqueous fraction of the product by various types of bonding and is not removed by sublimation GNgoddy, 1969). The specific amount of adsorbed moisture in equilibrium with various temperature and relative humidity conditions was required for accurate computer simulation of atmospheric freeze-drying. The experimental apparatus described in the following paragraphs was assembled to obtain the necessary equilibrium moisture isotherm data. The experimental device, shown schematically in Figure 4.3, was capable of subjecting a sample of freeze-dried beef to an atmo- sphere of pure water vapor over ice at controlled temperatures. The device consisted of a modified mass sorption Spring balance (Worden Quartz Products, Inc., Model 4401) connected through a 51 w to temperature recorder \\ _l ‘ a N} to temperature dt—- to vacuum pump control bath quartz spring U-tube manometer vapor condensor [3; sample 1 U 91%;: U —-4 [11:23 cross-hair —""""' +- I microscope KT I :24— ’I‘ I - from lower temperature control bath cooling jacket J l “a; Figure 4.3. Schematic Drawing of Experimental Apparatus for Equilibrium Moisture Content Studies. 52 vapor condensor to a vacuum pump (Welch, Model 1400). The vapor condensor was cooled with a mixture of solid carbon dioxide and acetone to prevent water vapor from entering the vacuum pump. The freeze-dried beef sample was mounted on a nichrome wire hook suspended on a quartz Spring inside the upper jacketed part of the vertical glass test cylinder. The upper jacket was connected by Tygon tubing to a heat exchanger located in a constant temperature chamber. The coolant, fifty percent ethylene glycol-water solution, was pumped through the jacket and heat exchanger by a 1/lS-horesepower centrifugal pump. The lower jacket, surrounding the ice, was controlled separately by a constant temperature bath (American Instrument Co., Model 4-8600) containing a built-in 1/30-horsepower pump. The upper end of the vertical cylinder was connected to a mercury manometer. The top of the mass sorption balance was sealed with a glass cap through which four tungsten probes had been placed to facilitate reading thermocouples inside the cylinder. Capper-constantan thermocouples were located near the sample in the area surrounded by the upper temperature control jacket and on the ice surface near the bottom of the cylinder. Thermocouples were read alternately on 30-second intervals by a recording potentiometer (Brown Division, Honeywell Inc., Model 153X65P12-X-2F). Temperature range of the recorder was -400C to 60°C with a minimum readable division of approximately .200. The temperature control equipment could control both jacketed areas to approximately : .30C.of the reSpective settings. 53 The amount of moisture adsorbed or desorbed by the sample was determined from the position of a cross-hair on a quartz fiber which was suspended below the sample so the cross-hair was visible between the two temperature control jackets. Position of the cross-hair was determined by sighting through the glass cylinder wall with a 10X ocular microscope (Nikken, No. 39837) with a readability of .01 mm. The quartz spring had an extension constant of .0243 gms/mm with a maximum load of five grams. Approximately .2 gm samples of freeze-dried beef were mounted on the nichrome hook. The stopcock and upper cap were lightly coated with Apezdon vacuum seal grease and turned into place. The cylinder was evacuated with the vacuum pump until the manometer recorded only the vapor pressure of ice at the prevail- ing temperature. The stopcock was then turned isolating the cylinder. Final adjustments were made on the two temperature control units to reach the desired condition, and the system was allowed to equilibrate. For a given isotherm the upper jacket temperature (which controlled the sample temperature) was left unchanged at the isotherm temperature. The lower jacket was set at a temperature correSponding to the desired vapor pressure, the sample was allowed to equilibrate, and a reading of spring deflection was recorded. The lower jacket temperature setting was then changed to correSpond to a new vapor pressure, and the process was repeated. Only desorption equilibrium moisture data were required, so that sample was first equilibrated to saturated vapor pressure conditions. The relative humidity of the water vapor surrounding 54 the sample was then lowered stepwise to give approximately seven data points over the entire relative humidity range. The lowest ice temperature which could be attained was -26°C. This temperature established the correSponding lower limit to the water vapor pres- sure which could be reached. Experimental Apparatus and Procedures: Rate Studies AtmOSpheric freeze-drying rate studies were conducted in a modification of the apparatus described in the previous section. A schematic drawing on the modified apparatus is shown in Figure 4.4. The lower end of the vertical cylinder was extended approx- imately 60 cm to assist in obtaining laminar flow of air over the sample. Also, the lower cooling jacket was removed, and a cartesian manostat (Manostat Corp., Model 7A) was installed between the vapor condensor and the vacuum pump. The manostat controlled the total pressure in the cylinder to approximately plus or minus one milli- meter Hg of the desired value when operating at pressures below one atmosphere. Air was circulated upward through the test chamber, then through a 6-cm by 25-cm cylinder of silica gel to remove water vapor from the air. From the desiccator air was pumped through a heat exchanger submerged in the Amico constant temperature bath and back to the lower end of the vertical cylinder. The heat exchanger consisted of fifty feet of 3/8-inch I.D. c0pper tubing. Connections between the various components were made with 3/8~inch thickewalled Tygon tubing clamped at each end. The entire air circuit excluding the heat exchanger was insulated with 3/8-inch thick refrigeration insulation (Armstrong, Armaflex). 55 to vacuum pump 1 to temperature recorder ’9 r .... manostat fl \ 1 cooling 3‘ 8 Mt r‘ to j ° U-tube manometer vapor condensor , (Ef’ Figure 4.4. Schematic Drawing of Experimental Apparatus for Atmospheric Freeze-Drying Rate Studies 56 The sample holder used in the rate studies is shown sche- matically in Figure 4.5. Due to sensitivity of the spring it was impossible to attach thermocouples to the sample being weighed. For this reason two sample holders, oriented one above the other, were used. The lower sample holder was suSpended from the upper holder by the quartz spring. Surface temperature and temperature of the frozen core were monitored by fine wire copper-constantan thermocouples in the upper sample. The upper sample holder was attached rigidly to a glass rod connected to the cap which sealed the upper end of the test chamber. Air temperature was measured just below the upper sample holder. Surface temperature, ice-core temperature, and air temperature were recorded in order on 30- second intervals using the same recording potentiometer described in the previous section. AtmOSpheric freeze-drying tests were conducted in samples of the shape of a finite cylinder cut from frozen beef cubes des- cribed in an earlier section of this chapter. An eight-millimeter diameter sharpened cork cutter was used to cut finite cylinders from the precut cubes. The radial surface of the cylindrical sample was sealed with saran film glued to the sample with Duco cement. The cylindrical sample was then mounted in a styrafoam sample holder as shown in Figure 4.5 such that the radial surface was thermally insulated. Thickness of the sample was trimmed to match the holder thickness (approximately one cm ). Heat and mass tranSport were effectively one dimensional through the ends of the sample cylinder. Natural grain of the meat was oriented either parallel to or per- pendicular to the ends of the cylinder at the time of sample glass support ing rod j—Q thermocouples \ temperature sample upper sample holder ~ ——-— test cylinder quartz spring\ ...—— lower sample holder weighted sample \ L l / 7 J7] / /J_4 / / l/ / / 1 / / / ‘ \ :“I \ N u ‘ \ \ \ [Z f/ / / 1V / f/ / f/ / / / / / y /l Figure 4.5. Sample Holders, Spring and Thermocouple Assembly for Drying Rate Studies. 58 preparation as required by the particular test conditions. Effectiveness of the vapor seal in preventing radial mass transfer was tested by atmOSpherically freeze-drying several samples approximately one-half way through the complete process. The samples were removed from the holder and the vapor seal was removed. The ice-vapor interface was distinctly visible on the radial surface indicating little or no sublimation had taken place from this surface. Prior to placing the samples in the test chamber the system was equilibrated to the desired test temperature. At the start of each test the silica gel in the desiccation cylinder was replaced. Used silica gel was regenerated by being placed in a drying oven at 100°C for not less than three days. Sample weight was monitored throughout a test by measuring extension of the quartz spring with the same microsc0pe as pre- viously described. Spring deflections were measured to the nearest .05 mm during drying rate tests. Deflection readings were taken on intervals of from one-half to two hours such that 15 to 50 readings were obtained per test. The spring extension constant was .0243 gm/mm. Accuracy of the weight recorded from spring de- flections was checked by weighing the sample before and after each drying test on a sample balance (Mettler, Serial no. 222912). Test conditions and results in the form of dimensionless weight versus dimensionless time are presented in Appendix V. Air flow rate was measured using a vertical tube flow- meter inserted in the air circuit between the pump and the heat exchanger. Mean air velocity over the sample was computed to be 59 approximately .45 meter/sec. Because of excessive air pressure drop across the flow meter the meter was removed from the air circuit during a test. Therefore, air flow during a test was somewhat greater than .45 meter/sec. Zero moisture content in any biological product is dif- ficult to define and much harder to measure. This difficulty is caused by various types of bonding between the moisture and other components of the product. Throughout the current research this problem was circumvented by defining the dry, moisture-free, state to be that level of moisture content reached by freeze-dried beef in equilibrium with silica gel at -l7.8°C. All test samples were equilibrated to this moisture content after atmOSpheric freeze- drying. The defined zero moisture content correSponds approximately to the same level reached by drying oven determinations, but does not damage the sample by subjecting it to high temperatures for extended periods of time. CHAPTER V THE NUMERICAL SOLUTION The nature of numerical solutions of complex mathematical models is such that the investigator faces numerous decisions on points of competition between completeness, accuracy, stability, and computation time of the model. The various compromises and decisions made with regard to the preposed model are discussed in this chapter. The numerical representation of the mathematical model derived in Chapter III was solved with computer subroutine MODEL listed in Appendix IV. The computer program required explicit evaluation of several functional relationships and physical con- stants which appear in the model. These evaluations are discussed in this chapter. Last, qualitative results are presented to support the hypothesis that the proposed model is an adequate representation of atmOSpheric freeze-drying. The most important functional relationship required for the numerical solution was the equilibrium moisture content of freeze- dried beef. The mathematical model, as derived in Chapter III, assumed adsorbed moisture in the porous zone was in equilibrium with the air-water vapor mixture in the pores. Equilibrium adsorbed moisture in freeze-dried beef was determined using the apparatus and procedures described in Chapter IV. Results of these tests are presented in Figure 5.1 as a function of the relative humidity of 60 61 the water vapor. Other investigators (Saravacos and Stinchfield, 1965) have obtained adsorption isotherms in freeze-dried beef for temperatures below zero oC. Adsorption data by these investigators are presented in Figure 5.1 as a dashed line. It is of particular interest to note that equilibrium moisture contents obtained in separate investigations are in excellent agreement at the saturation condition. Adsorption and desorption data are not expected to agree over all of the relative humidity range due to sorption hystersis commonly found in bio- logical products. As demonstrated by results presented in Figure 5.1, variations in equilibrium moisture content with temperature below zero oC.are small. Advantage was taken of this fact in derivation of the model. The variation in equilibrium moisture content with reSpect to temperature was considered negligible when compared to variation with respect to relative humidity of the water vapor. The derivative of equilibrium moisture content with reSpect to relative humidity was required in the mathematical model. In- Spection of Figure 5.1 shows the average value of the derivative for the desorption isotherm to be approximately .2. That is, AM/Ar 7 .2. Deviation from this average value is significant above a relative humidity of .5; however, attempts to fit the desorption isotherm or its derivative with various expressions failed to pro- duce sufficiently accurate results to allow a stable numerical solution of the model. Therefore, the derivative of equilibrium moisture content with reSpect to relative humidity was approximated as the mean value of the derivative, namely, .2. M - Adsorbed Moisture Content, dry basis 62 ‘3 -6.7OC, desorption, cooked freeze-dried beef ‘5 -10.OOC, adsorption, raw freeze-dried beef E] -20.0°C, adsorption, raw freeze-drief beef (adsorption data by Saravacos and Stinchfield, 1965) .2 '1 ”’9 00 o B 1 / )I 4 I 2°" * / / / 3 / I / .1 .. ’E 1’ I 1 + 0 I I I fit I .0 .2 .4 .6 .8 1.0 r = P /P V sat Figure 5.1. Equilibrium Moisture Isotherms of Freeze- Dried Beef Below Zero 00. 63 The total amount of adsorbed moisture present in the porous layer was important to accurate analysis of the freeze-drying pro- cess. Previous mathematical models of freeze-drying have neglected adsorbed moisture altogether (Sandall, g£_al., 1967; Dyer and Sunderland, 1968). Figure 5.1 shows that approximately .2 gm-HZO/ gm-solid exists in the porous zone in equilibrium with the ice interface. This is approximately 10-15 percent of the initial moisture content. In addition, the amount of heat required for desorption of this moisture increases as the moisture content de- creases. Ngoddy (1969) has evaluated the heat of sorption for freeze-dried beef as a function of adsorbed moisture content. Above an adsorbed moisture content of approximately .2 gm-HZO/ gm-solid the heat of sorption is nearly constant at the value for free water vaporization. Below that moisture level the required heat of vaporization increases rapidly due to increasingly stronger bonding of the remaining water to nonaqueous components of the product. Since a gradient of adsorbed moisture remained in the porous zone when sublimation of free moisture was complete, ne- glecting the adsorbed moisture could be expected to have significant effect on the total drying time predicted by the model. The saturated pressure of water vapor as a function of temperature was also required in the numerical solution. This expression was obtained by fitting a third degree polynomial to saturated vapor pressure data over the temperature range from ~3000 to zero oC. Values of the vapor pressure over ice were obtained from Threlkeld (1962). Coefficients of the polynomial were evaluated by minimizing the sum of squared deviations from 64 the data. Maximum absolute deviation of the polynomial was within three percent of the data over the temperature range of interest. Various physical parameters which were assumed constant in the model also required evaluation. The constant numerical values used in the solution and the source of information are listed in Table 5.1. The computer solution required the sample surface temperature as a known input. This temperature as well as the ice core temper- ature and air temperature were monitored during tests performed to collect parameter estimation data. For atmOSpheric freeze-drying it was found that the surface temperature remained nearly constant within one-half 0C of the air temperature. Therefore, surface tem- perature was entered in the computer program as a constant value. TABLE 5.1. Numerical Values of Physical Constants Used in the Mathematical Model Constant Value Source Bulk density of dry .46 gm/cm3 Mean value of experimental product, p measurements Specific heat of dry .38 cal/gm-OC Computed from specific product, C d heat of frozen beef at P -4o°c (Short and Staph, 1951) Specific heat of ice 1.15 Cal/gm-OC Riedel (1957) core, C pc Porosity of dry .76 Harper (1962) product, e Mutual diffusivity, air .22 cmZ/sec Perry (1963) and water vapor, 1 atm, zero C, D Heat of sublimation, AHS 676 cal/gm Threlkeld (1962) Initial moisture approx, 1.5 Measured for each test content, Mo d.b. Half-thickness of approx. .45 cm Measured for each test sample, 3 65 If the computer subroutine, MODEL, were used to analyze data obtained in vacuum freeze-drying it would be expected that the surface temperature would vary with time. In this case values of surface temperature experimentally measured on some time in- crement could be stored in common storage by program MAIN much the same as experimental values of dimensionless mean moisture content and time were stored (See Appendix IV for listing of program MAIN). The surface temperatures could be used in subroutine MODEL as a boundary condition on the energy equation. Solution of finite-difference equations to accurately approximate the partial differential equations from which they were written required careful selection of the incremental step size in the independent variables. There are two independent variables in the proposed model: dimensionless time and dimension- less distance. Size of the increment in dimensionless distance, A®, was selected by repeatedly solving the model while varying the size of the increment. An increment of .1 gave three significant figures in the dependent variable (dimensionless mean moisture content) when compared to the solution obtained with an increment of .05. This level of accuracy was equal to the accuracy of ex- perimentally determined dependent variables, so a distance incre- ment of .1 was selected. The maximum time step, A9, compatible with stability of the convective mass transfer boundary condition was computed internally in computer subroutine MODEL using the inequality of equation (3.23). The computed time increment was approximately 40 seconds, real time. Computation time for one step in real time on the CDC-3600 computer 66 varied depending on the number of nodes in the numerical solution at any given time. Average simulation speed for solving the entire model was approximately 60 hours per minute of computer time. Difficulty was encountered in finding a stable numerical representation of the energy equation, equation (3.1). As mentioned previously, application of the analytic stability analysis developed by Von Neumann (See Smith, 1965) to the energy equation indicated either the Crank-Nicolson or backward-difference approximation method should have been stable. This method of stability analysis is based on expressing an error from the correct solution in terms of a Fourier series. If the series converges the numerical approx- imation is stable. In application of the method the coefficients and domain of the equation in question are considered to be constant. In Spite of the successful stability analysis of equation (3.1) trial solution of the proposed model revealed the energy equation was unstable. Use of the backward-difference method provided a stable solution longer than the CrankrNicolson approximation; however, both methods eventually became unstable. The inconsistency between the stability analysis and actual solution was apparently explained by nonconstant coefficients and the expanding domain of the model. Further trial solutions served to confirm that no feasible combination of space and time increments could maintain stability in the energy equation indefinitely. It was also established that instability in the energy equation was directly associated with the term accounting for the heat of vaporization of moisture being de- sorbed in the porous zone. This so-called sink term was time 67 dependent since the rate of moisture desorption at any point in the porous zone was a function of time. A stable representation of the energy equation was finally obtained by neglecting the energy re- quirement associated with the desorption of moisture in the porous zone. The amount of adsorbed moisture present was still accounted for in the mass transfer equation. The final energy and mass equa- tions of the mathematical model which were solved simultaneously for the numerical solution are given in equations (5.1) and (5.2) reSpectively. D 5P 5: = a_ E 51) .9 C V BE. c + c + W— 5.1 p( pd M pw)ae ab 56 D M acb ( ) M P P [ii+—9_ ®h=L£EJJ (5.2) RT Psat as M a¢ Qualitatively the effect of disregarding the heat of vapor- ization for adsorbed moisture can be viewed as removing one of the requirements for the heat being transferred from the surface to the ice-vapor interface. The net result was that the model slightly overestimated the heat flux to the interface. Overestimating the heat flux to the interface caused the core temperature to be over- estimated and a correspondingly higher vapor pressure at the inter- face to be computed. The higher vapor pressure in turn caused the sublimation rate to be slightly overestimated. Quantitatively, the error caused by neglecting the heat sink term in the energy equation was small. The total heat required to sublimate the free moisture from a unit weight of frozen beef was approximately ten times the heat required to vaporize adsorbed moisture. 68 Finally, some finite initial domain was required in order to write the finite-difference equations for heat and mass transfer in the porous zone. An initial domain was generated by assuming heat and water vapor were exchanged between the air stream and an exposed ice surface while the core temperature dropped from the initial temperature to the wet-bulb temperature of the air stream. This time was observed to be approximately five minutes in most tests. The initial domain was approximately two percent of the sample half-thickness. Initially the numerical solution contained only two nodes: one on the surface and the floating node on the ice-vapor interface (See Figure 3.1). As the ice-vapor interface receded more nodes entered the solution. Trial solutions of the numerical model gave indication that the model was at least qualitatively correct. Figure 5.2 shows typical vapor pressure and temperature profiles computed from the pr0posed model for one-dimensional atmOSpheric freeze-drying of beef. For the elapsed time shown in Figure 5.2 three-fourths of the free moisture had been removed. The position of the ice-vapor interface is indicated by a dashed line. The most outstanding characteristic of the computed profiles is their almost perfect linearity. These computed results strongly supported the pseudo steady-state assumption used by previous investigators (Sandall, fig 1., 1967; Dyer and Sunderland, 1968). Clearly, movement of the ice front was so slow that the time derivatives of dependent variables in the porous zone were insignificant compared to the space derivatives. '20 -3. U 0 I Q) .. u 4. :1 U m L: g-S. E (D E-I ' '60 [-I 4; an O H M “ii Q, 3. ..C G. U) 0 E U (U I a 2. :3 03 U) £3 I4 0 D. (U > 1. I-o Q) J.) N 3 I 04> 0 4 . I . l l . I I I I ‘ I l I i I I I . , . . . T . . . j .0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 ¢ - Dimensionless Position Computed Profiles of Vapor Pressure and Temperature for Typical Conditions of Atmospheric Freeze-Drying. Figure 5.2. 7O Linearity of the computed dependent variable profiles shown in Figure 5.2 provided increased confidence in the accuracy of the computer solution early in the process when only a few nodes were in the solution. If the profiles of the dependent variables were highly nonlinear, accuracy of the solution would have been re- stricted during the early portion of the process when the number of active nodes was small. Linearity of the profiles also increased the accuracy with ‘whiCh the derivatives of vapor pressure and temperature were computed at the interface. The velocity of the interface and the rate of drying were dependent on calculation of these derivatives. Figure 5.3 shows a solution of the proposed model in terms (of dimensionless mean moisture content versus dimensionless time. 'The model solution has been converged to a typical set of atmOSpheric freeze-drying data obtained from a one-dimensional sample. Also sshown in Figure 5.3 is a solution which neglected the same compo- tlents of the heat and mass transfer equations as did the pseudo Eiteady-state model. The most important component neglected was I:he adsorbed moisture. Both solutions included the same values for éill constants and tranSport parameters of the model. The proposed tnodel, at least qualitatively, predicted the extended time required t:o remove the adsorbed moisture after the ice front reached the <2enterline. Previous investigators (Sandall, _E _l., 1967) have Iloted that models which neglected the adsorbed moisture were Eiignificantly in error after 75 to 90 percent of the original tnoisture had been removed. This observation was supported by results of the proposed model. 7l .oe .mmHQEmm HmcofimCmEHQumco 5H Hopoz mumum umcmmum opommm paw proz pom0doum msu mo mo>uso w5w>ua pouofipmum mo somwumasoo .m.m munwwm o u mEHH mmmaconCmEHQ .on .oN .oa - b. p - Hobo: oumumumvwmum ousomm Hobo: pmmOQoum o.a u ‘nuaquog aannsrow uean ssaIuoisuamiq 72 Combined results shown in Figures 5.2 and 5.3 indicated the pseudo steady-state model was in error chiefly due to neglecting the adsorbed moisture in the porous zone. Clearly the assumption of linear dependent variable profiles was acceptable. However, for purposes of parameter estimation, the more complete description of atmospheric freeze-drying as included in the proposed model was preferred. Qualitative results and discussion presented in the pre— ceding paragraphs indicate the proposed model is probably an adequate representation of atmospheric freeze-drying. More quan- titative and strenuous tests of the model will be discussed in the next chapter under the headings of parameter estimation and analysis of residuals. CHAPTER VI RESUETS AND DISCUSSION Three tranSport parameters of the mathematical model were evaluated from data generated in atmospheric freeze-drying tests described in Chapter IV. Estimates of these physical parameters are presented and discussed in the first section of this chapter. The following section includes analysis of the residuals and further discussion of the validity of the one-dimensional mathe- matical model. In subsequent sections of the chapter the proven model is transformed into an approximate, three-dimensional model and used for practical analysis of the effect of all Operating variables on the rate of atmOSpheric freeze-drying. The parameter estimates Three tranSport parameters were evaluated in each of 18 tests conducted at six different experimental conditions. At .97 atm total pressure, tests were conducted at all four possible combinations of air temperature (-2.80C and -8.20C) with fiber orientation (parallel and perpendicular to the direction of trans- port). In order to adequately test the accuracy of the mathematical model, data were also taken at a system pressure of .58 atm and -2.8OC air temperature for both parallel and perpendicular fiber orientation. 73 74 Before presenting the parameter estimation results a brief diversion is in order to explain the statistical information obtained with the parameter estimates when the GAUSHAUS program (Meeter, 1964) was used for nonlinear estimation. As mentioned previously, GAUSHAUS is a library program which utilizes a combination of methods to perform estimation of parameters in mathematical models which are nonlinear with respect to their parameters. This program was used chiefly because of the supplementary information which was obtained with the parameter estimates at little or no extra effort. This supplementary statistical information helped to evaluate the accuracy of the model and the parameter estimates; it included: 1. final functional or predicted values of the model using the optimum parameter estimates. 2. approximate 95 percent confidence limits on the pre- dicted functional values and on the parameter estimates. The confidence limits were computed from a linear approximation of the model in the vicinity of the optimum parameter matrix, Q, and, therefore, were not exact. 3. residual values, that is, (Yi - Mi) for i = 1,2, . . ., L. These values are analyzed in the next section of this chapter to assess the accuracy of the model. 4. variance of the residuals which, in the case of a linear model, is an independent and unbiased estimate of 02, the variance of the individual observations. In the nonlinear model the estimate is biased but can be used as a relative measure of the variance of observations between tests. 5. the correlation matrix, which revealed how the various parameters were correlated with each other. This Supplementary information is mentioned in the following dis- cussion of the model and the parameter estimates. Eighteen estimates each of three parameters are presented in Table 6.1. Initial observation indicated substantial variability in 75 all of the parameters. This was probably to be expected in a bio- logical product where composition can vary from sample to sample. A more encouraging observation was that the internal transport parameters, k and C2, were estimated near the values expected. Harper (1962) reported the value of k in freeze-dried beef for pressures above approximately .5 atm in the absence of a counter- flow of water vapor to be approximately 1.5x10-4 cal/cm-sec-OC. The estimates of k shown in Table 6.1 are near this value. Sandall, £5 31. (1967) evaluated CZ, the structural constant in the effective mass transfer coefficient, in the breast meat of turkey. They found C to be between .44 and .66 for tranSport 2 parallel to the fibers and approximately .27 for transport per- pendicular to the fiber orientation. Similiar to slightly higher values are presented in Table 6.1 for precooked beef. A summary of the analysis of variance of all three para- meters is presented in Table 6.2. The variance of the experi- mental results was analyzed for significance due to air temperature, system pressure and orientation of the fibers. Interactions be- tween these factors were assumed negligible. Testing for significant differences was done by use of the F-test at the 90% level of significance (Peng, 1967). 76 TABLE 6.1. Summary of Parameter Estimates and Variance of the Residuals Test 0 h c k 2 n ' D 2 0 Air Temperature -8.2°C Pressure = .97 atm Orientation = parallel 1 .0084 .56 .46x10-4 1.97x10'S 17 .0091 .75 .44 9.78 2 .0079 .86 .47 5.72 Air Temperature -8.20C Pressure = .97 atm Orientation = perpendicular 7 .0087 .64 2.77x10'4 1.05::10'S 8 .0117 .80 .95 29.90 18 .0086 .51 1.59 13.80 Air Temperature -2.8OC Pressure = .97 atm Orientation = parallel 3 .0126 .89 .53x10'“ 21.82:.10'5 4 .0116 .95 1.35 1.97 15 .0056 .99 .51 16.44 16 .0092 .61 .31 5.36 Air Temperature -2.80C Pressure = .97 atm Orientation = perpendicular s .0072 .62 .96x10'4 9.24x10’5 6 .0111 .64 .99 17.26 Air Temperature -2.80C Pressure = .58 atm Orientation = parallel 9 .0087 .74 1.21x10‘4 8.5le0’5 10 .0087 .99 1.00 12.36 12 .0095 .74 1.02 7.54 Air Temperature -2.8°C Pressure = .58 atm Orientation = perpendicular 11 .0133 .42 1.03x1o'4 14.12x1o'5 13 .0091 .72 1.75 8.76 14 .0107 .62 1.09 11.40 77 TABLE 6.2. Summary of Analysis of Variance in the Parameter Estimates Analysis of Variance in Estimates of hD Source 2£_Variance d.f. Mean Square F-ratio vs F(.9Q,l4,l) Air Temperature 1 2.007 .425 3.10 System Pressure 1 1.910 .405 3.10 Orientation of Fibers 1 3.759 .897 3.10 Experimental Error 14 4.717 Analysis of Variance in Estimates of C 2 Source 2£_Variance d.f. Mean Square F-ratio vs F(.90,14,l) Air Temperature 1 1.395 .681 3.10 System Pressure 1 .319 .156 3.10 Orientation of Fibers 1 15.855 7.755 3.10 Experimental Error 14 2.045 Analysis of Variance in Estimates of k Source gf‘Variance d.f. Mean Square F-ratio vs F(,90,14,11 Air Temperature 1 .651 .231 3.10 System Pressure 1 6.264 2.224 3.10 Orientation of Fibers 1 7.978 2.76 3.10 Experimental Error 14 2.822 78 Turning attention now to analysis of each parameter estimate, the surface mass transfer coefficient, hD, is considered first. The value of this parameter was not expected to vary with temperature I or pressure over the small range of these variables that was con- sidered. Since hD is not a function of product properties it was not expected to be a function of orientation of the fibers. Statis- tical analysis of the dependence of hD on temperature, pressure and orientation as summarized in Table 6.2.confirmed that no significant difference exist for any of these factors at the 90 percent confidence level. The mean value of all estimates of hD was approximately .0095 gm/sec-cmZ-atm. The internal heat and mass transfer parameters, being functions of the product under consideration, were of greater inter- est. The structural constant in the effective mass transfer co- efficient, C2, can be viewed as an attenuation constant which accounted for the amount the free-gas value of the mutual diffu- sivity of air and water vapor was reduced due to constrictions of the porous media. Krischer (1959) has related this constant to the porosity of the porous zone by a factor to account for the tortuosity of the path of the water vapor molecule through the dried portion of the product. (6.1) The porosity of freeze-dried beef has been reported by Harper (1962) to be approximately .76. Since it was expected that the tortuosity factor, T, was greater for tranSport perpendicular to the fiber orientation of the meat than parallel to the fibers, C was expected 2 79 to be less in those tests conducted with water vapor tranSport perpendicular to the fibers of the meat. The structural constant was not expected to be a function of any operating variable. The estimates of C2 shown in Table 6.1 were tested for significant differences due to temperature, pressure level and orientation of the fibers. Results of this analysis of variance are presented in Table 6.2. Only differences in C2 due to orientation of the fibers were significant when tested at the 90 percent confidence level. Differences due to fiber orientation were also significant at the 95 percent confidence level. The mean for estimates of C2 for vapor diffusion parallel to the fibers was .81 and .62 for diffusion perpendicular to the fibers. Estimates of the effective thermal conductivity, k, pre- sented an interesting comparison to results obtained by Harper (1962). Using steady-state methods on freeze-dried beef with no water vapor flux, Harper found the mean value of k to be 1.5x10-4 cal/cm-sec-OC. An overall mean value of 1.0x10-4 cal/ cm-sec-OC was found in the current research. These estimates were made in the presence of a counterflow of water vapor and by parameter estimation from transient experiments. The estimated effective thermal conductivity was especially sensitive to varia- tions in structure and composition of the meat sample as is evident from the results of Table 6.1. In addition, an unknown portion of the total variation in the estimated values of k can be attributed to the elongated contours of the risk-function surface in the k direction (See Figure 4.2). Ninety-five percent confidence limits were computed for the estimated value of k. Using the t-test 80 (Snedecor, 1956) these limits were found to be .7x10-4 to 1.3x10-4 cal/cm-sec-OC. Variations in k due to system pressure, air tem- perature and orientation were all insignificant at the 90% confidence level. It may be argued that more powerful techniques are avail- able for obtaining the single best estimate of the parameter matrix than by finding the arithmetic mean of each parameter individually. If the nonlinear model were represented with a linear approximation in the vicinity of the optimum parameter matrix of all the tests, the risk function would be an (n)-dimensiona1 parabaloid over the domain of the parameter matrix, where (n) is the number of parameters estimated. The single best estimate of 8. could then be found by finding the minimum of the parabaloid. However, linearization of the nonlinear model can only be accomplished over incremental variations in the estimated parameters. It was concluded that the variation shown in the parameter estimates of Table 6.1 could hardly be construed to be incremental in magnitude. Therefore, the arith- metic mean value was computed to be the single best estimate of each parameter. Evaluation of k and C2 allowed certain observations to be made concerning details of the mechanisms of atmospheric freeze- drying. Heat transfer through the porous zone of the product has been assumed to be by conduction through the solid matrix and by some combination of conduction and convection through the gas- filled pores. The mean of the current estimates of R was approx- imately two-thirds of the magnitude which Harper (1962) found. While variability of the estimates of R was large it is noteworthy 81 that the 95 percent confidence limits on the mean of the current estimates did not include the mean value Harper obtained. Further- more, the mean estimated value of k was between the values Harper found for atmOSpheric pressure and vacuum conditions (See Chapter II). These points tend to support the above concept of the mechanism of heat transfer with some additional insight. Apparently, the counter-flux of water vapor throughout the drying process sub- stantially reduced the contribution to the pores to transfer of heat in the opposite direction. Thus the effective value of thermal conductivity measured under dynamic conditions at atmospheric pres- sure was found to be near the value obtained under static conditions in a vacuum. Implication of the above results is that the effective transfer of heat through the porous zone during the drying process is substantially less than measured under steady-state conditions. Such findings are important to optimization of the freeze-drying process. The rate of atmospheric freeze-drying has been observed to increase significantly with increasing air temperature in the range of 4100C to zero OC. The question has been raised as to whether this phenomenon was partially caused by liquid tranSport of water which was unfrozen due to the presence of solutes. Tests were conducted in the current research in an attempt to answer this question. The maximum temperature at which ice exists in frozen beef is approximately -1.750C (Hohner and Heldman, 1970). Experimental atmospheric freeze-drying tests were conducted at air temperature 82 of -2.80C and -8.20C. If significant liquid transport had resulted in.tests at the higher airtemperature,the accelerated drying rate would have been reflected in an inflated estimate of the mass trans- fer parameter in these tests. Statistical analysis of estimates of C2 summarized in Table 6.2 indicates the mean value from tests at -2.800 was slightly but not significantly larger than values from the lower temperature tests when tested at the 90 percent confidence level. Thus the concept of liquid tranSport was not supported by the results. The frozen core temperature was observed experimentally to remain three to five degrees C below the air temperature throughout most of the drying process. This observation was confirmed by the solution of the mathematical model. In summary, it appears unlikely that water was transferred in the liquid state when the air temperature remained below the initial freezing point. The fraction of product moisture in the form of ice increases rapidly with decreasing temperature in the range just below the initial freezing point; thus depression of the ice core temperature served to prevent liquid tranSport of water. In a later section of this chapter it is demonstrated that the proposed model does predict the observed in- crease in the drying rate with increasing air temperature. Results of statistical analysis of the mass transfer para- meter, CZ’ failed to reject the hypothesis that the mechanism of mass transfer was water vapor diffusion through stagnant air in the pores of the dried layer. Further insight into this process was gained from the magnitude of the estimated values of C both 2 parallel and perpendicular to the fiber orientation. The mean 83 value of C2 parallel to the fibers was determined to be .81 compared to a value of one under free-gas conditions. In other words, the mean free path of the water vapor molecule led to contact with the porous solid only often enough to reduce the effective value of the free-gas mass diffusivity by 19 percent. Similarily transport perpendicular to the fibers was reduced by 38 percent. From the viewpoint of a water vapor molecule at freeze-drying temperatures, freeze-dried beef is a highly porous medium. The fact that Sandall gt El- (1967) estimated C2 to be between .44 and .66 parallel to the fibers and .27 perpendicular to the fibers of turkey meat may have been because the structure of turkey meat is less porous than that of beef. However, these lower values for C2 may also have been computed due to fitting the incomplete pseudo steady-state model to experimental freeze- drying data. From Figure 5.3 it can be observed that a lower mass transfer coefficient would have been required to cause the pseudo steady-state solution to fit the same data that the model used in this research fit with a higher value. Accuracy of Ehg_Mode1 Qualitative results have been presented in Chapter V to support the accuracy of the numerical solution of the model and to compare it to previous models. Comparison of model solutions to experimental results and analysis of the residuals between the experimental data and the computed values are presented in this section to further confirm accuracy of the model. 84 Figure 6.1 compares the solution of the model to experi- mental results of one-dimensional tranSport tests at two different pressure levels. The computed solution represents the optimum fit of the model to each separate set of data. Ability of the model to fit results obtained at different levels of the operating vari- ables is further demonstration of the accuracy of the numerical solution of the model. Results of Figure 6.1 indicate the model satisfactorily fits each set of experimental results. The difference between the final functional value of the model and the experimental value at each recorded point is called the residual value. Draper and Smith (1966) discussed several methods of analyzing the size, randomness and various trends which the residual values may exhibit. If the mathematical model were a complete and accurate representation of the physical process under study and all experimental data were obtained with an un- biased procedure the residual values of any test would be a random variable with magnitude equal to the standard deviation of the experimental error. The objective of the methods of analysis presented by Draper and Smith (1966) was to answer the question whether the residual values have the characteristics of a random variable. The residual values of all 18 one-dimensional atmOSpheric freeze-drying tests used to obtain parameter estimation data are shown in Appendix V. Variance of the residuals of all tests is shown in Table 6.1. In all cases the residuals were small, almost never greater than two percent of the initial functional value. In addition, the variance of the residuals was small and quite Dimensionless Mean Moisture Content, M 1.0 08¢ .2 a? 1w l 0 .0 I . 10. Figure 6.1. 85 0 - .97 Atmosphere Total Pressure 0 - .58 Atmosphere Total Pressure I If 1* I 20. 30. 40. 50. Dimensionless Time - 9 Comparison of the Proposed Model to Experi- mental Results at .97 and .58 Atmosphere "l‘nf-n'l Drpnnnrn _ 86 uniform over the entire group of tests. Average variance of the residuals was approximately lxlO-4. InSpection of the time dependence of the residual values as they are listed in Appendix V revealed a cyclic nature in all tests. No formal testing was required to confirm that the resid- uals were not a random variable. Such nonrandom cyclic patterns of residuals with relatively small values have been encountered before when tranSport parameters have been estimated from numerical solutions of mathematical models (Beck, 1969B). The small size of the residual values and the rather uniform variance of the residuals between tests tended to vindicate the experimental technique of inducing a biased error into the results.~ A more probable cause of the nonrandom residual values in all of the test results was the fact that several variables in the mathe- matical model were assumed constant. Assuming minor variables were constant tended to induce a small bias into the model. The parameter hD can be taken as an example. The surface mass trans- fer coefficient was assumed constant; however, being a minor func- tion of the water vapor concentration at the surface, this para- meter may have declined in value as the vapor concentration de- clined. The effect of such a variation can be seen from Figure 6.1. The experimental results initially declined faster than the model. Later when the value of hD had declined the experimental results fell more slowly than the solution of the model. Through the process of minimizing the squared deviations between the model solution and the data a mean value was found for parameters which actually were minor variables. The cyclic nature of the residuals 87 was caused by the model solution being based on the computer value of such parameters. The small size of the residuals in all tests indicated the total error in the proposed model was small; however, there was no means of computing exactly how accurate the model was. Since any addition would only increase the complexity of the model and its solution without insuring an improvement in the accuracy as measured by analysis of the residuals, a cost-benefits decision remained with the investigator. Therefore, the model was described as adequate for parameter estimation and process analysis, but probably not a complete description of the physical process represented. Simulation of Atmospheric Freeze-Drying in Three Dimensions In all tests discussed previously in this thesis the trans- port of heat and water vapor in the sample has been limited to one dimension. Such tests were used for parameter estimation and analysis of the mechanisms of atmOSpheric freeze-drying. Obviously, practical application of the process occurs in samples where heat and water vapor are transported in three dimensions. Complexity of the mathematical model would defy even numerical solution if it were derived initially in three Space coordinates. However, the one-dimensional model was transformed into a reasonably accurate approximation of atmOSpheric freeze-drying in cubical samples with tranSport of heat and water vapor from all six surfaces. Mean values of all parameters determined in the previous section were used, and the anisotrOpic effect induced by the fiber structure was disregarded. 88 Geometrically the three-dimensional model was visualized as a pyramid with height one-half the length of the base. The apex of the pyramid was located at the center of the cubical sample with the base of the pyramid on the sample surface. All tranSport of heat and water vapor was assumed to move perpendicular to the sample surface. Actually, of course, flow of heat and water vapor were not perpendicular to the surface of the cube except along a line from the center perpendicular to the surface. Nevertheless, con- sidering the sample variability reflected in the parameter estimates of a previous section and the effect of this variability on the product-dependent constants of the model, the three-dimensional model was considered sufficiently accurate for process analysis work. Results of the three-dimensional solution using the mean values of parameters are compared to experimental results of atmOSpheric freeze-drying of cubes of precooked beef in Figure 6.2. Both sets of experimental results Shown in Figure 6.2 were obtained at -2.80C air temperature and .97 atm total pressure. One sample had a half-thickness of .7 cm and the other .5 cm. These reSults confirm that the approximations included in the three- dimensional model were reasonably accurate until the dimensionless mean moisture content dropped below .1. At low moisture contents the three-dimensional model predicted excessively long drying times. At low moisture content the ice core in the three-dimen- sional model was assumed to be reduced to a small cube in the center of the sample. Water vapor was assumed to flow outward only along a path with cross-sectional area equal to the area of 89 .mmaaamm Hwownoo 5 muaomom Hmucmewuoaxm 3 Home: Hmcogcoeflnuoouna 65 mo mcoHuaaom mo comwumano mason .u 1 mega .N.8 868682 .ON .0@ .on .oe .om .ow .OH .0 hi - b, n b h. - III a. .0 It, 0 o o o o o o. .c on o o o o co. .0 o. O o. o o o womm omxoooopm .oHQEwm nouoEHucoo oé .. O o moon ooxooooum .3923 .8383ch a; n o on o o o.H w ‘nuanuoo BJnJSION ueaw 9331uoisuam1q 90 the ice core in the center of the sample. Clearly, near the end of the process, this assumption neglected a substantial amount of the effective tranSport area of the sample. In subsequent dis- cussion, where the three-dimensional model will be used for analysis of the atmospheric freeze-drying process, prediction of the model will be disregarded below MI= .1. Analysis of Atmospheric Freeze-Drying in Cubical Samples The power and economy of a proven computer simulation for analysis of the effect of operating variables upon a physical pro- cess quickly becomes apparent when the Speed and flexibility of the model solution are compared to acquiring the same information from experimental tests. The approximate three-dimensional model discussed above was used to investigate the effect of air tem- perature, system pressure, sample size, and magnitude of the Sur- face mass transfer coefficient on the rate of atmospheric freeze- drying in cubical samples of cooked beef. The practical operating range of all variables was in- vestigated by changing the variables one at a time while holding all others at a standard condition. The standard condition was the following: Air temperature, Ta = -3.00C System pressure, P = .97 atm Sample half-thickness, s = .5 cm Surface mass transfer coef., hD Structural constant, C Thermal conductivity, E Initial Moisture content, Mo .0095 gm/cm -sec-atm .725 o .0001 cal/cm-sec- C 1.5 gm-HZO/gm-dry solid Other product dependent constants were evaluated as shown in Table 5.1. 91 In Figure 6.3 the effect of air temperature on the rate of atmospheric freeze-drying of one centimeter cubes of cooked beef is shown. Air temperature was investigated at -3.OOC, -8.00C, and -13.0°C. This modest range in air temperature caused a greater change in the predicted drying time than the changes investigated in any other variable. The predicted increase in drying rate with air temperature was approximately of the order witnessed by Wood- ward (1961) and Lewin and Maletes (1962). As has previously been noted the practical upper limit of air temperature is approximately -3.0°C due to depression of the freezing point caused by dissolved solutes. Results of Figure 6.3 demonstrate that any practical application of atmOSpheric freeze-drying must be carefully designed to Operate as near the maximum allowable temperature as possible. The large dependence of drying rate on air temperature was caused by two factors. First, higher air temperature resulted in a higher ice core temperature which in turn caused a higher satu- rated vapor pressure at the ice-vapor interface. The higher vapor pressure at the interface represented an increase in the mass trans- fer potential and caused more rapid vapor tranSport across the porous zone. The second and minor cause was an increase in the vapor diffusivity due to increase in temperature. The effect of reducing system pressure on the rate of freeze- drying in one-centimeter cubes of precooked beef is shown in Figure 6.4. Little discussion is required concerning these results since it has been previously established that the maximum rate of freeze- drying occurs in the range of 8-25 mm Hg, far below atmospheric con- ditions. The justification of atmospheric freeze-drying centers 92 .wmmm nmxoooopm mo monso nonmewucooumco cw mmumm wcwzunimumoum opposamoEu< co musumumaEmH uw< mo uoowmm mouowooum .m.o muowfim musom .u a mafia .ow .05 .oo .om .oq .om .ON .OH .0 P p n P b P P - 00m 1 - a vow t coma- o z .udoucoo ououmfioz HmwuficH pHHOm muou8w\ommuew m.H o noomIEU\Hmo Hooo. u x .kuw>wuoomaoo HmEumLH o mmn. u Nu .ucmumcoo Honouosuum Eumnoomu Eo\8m mmoo. u 9: ..umoo powmcmuh mmmz oomwusm N Eu m. u m .mmmcxowneimamm oHQEmm Sum mm. u m .musmmoum Eoumzm o.H n ‘3ua3uoo BJHJSION ueaw ssaIuoisuamiq 93 wumm mewmuauouooum owumnamoeu< co whammmum Eoumxm mo uoomwm mouuwoonm .05 .oo .moom pmxooooum mo monso pmuwefiucoouoao cw wusom .u 1 mafia .0m .00 .om .ON oHHOm humusw\onqu m.H nommuEU oo \ EumlUQmINEU H66 Hooo. man. \am maoo. Eu m. coo m- 866 as. u a 886 me. u a EU“ m. "m 02 .ucoucoo ousumwoz HmHuHCH m .huw>auo:ocoo anemone o .ucwumcou Housuosuum as ..umou pommamua mmmz moomuom m .mmooxofisuanwm oemamm we .ousumuoaaoa uw< .q.o muowfim OOH T V Y n ‘uqaquoa axnasrou_uean $331uoxsuam1q 94 about elimination of equipment related to providing and maintain- ing a vacuum condition. However, process design for atmospheric freeze-drying should take full advantage of the increased drying rate due to reduced system pressure by arranging equipment to minimize the total pressure in the drying chamber. The effect of sample size on the rate of atmOSpheric freeze- drying in beef cubes at the standard conditions listed above is shown in Figure 6.5. Time to dry to M'= .1 decreased rapidly with decrease in the dimension of the cubical samples. For freeze- drying with a very large surface mass transfer coefficient (so that the surface vapor concentration approximates the free-stream vapor concentration) the time to dry to any given dimensionless moisture content should vary with the ratio of the square of the sample size. Results Shown in Figure 6.5 were computed with hD = .0095 gm/cmz- sec-atm which was not large enough to fulfill the above criterion. Nevertheless sample size was shown to greatly effect the drying rate. Effect of the surface mass transfer coefficient on the rate of atmospheric freeze-drying is best illustrated by definition of a ratio of external to internal mass transfer coefficients analogous to the heat transfer Biot number. H = hDs/De = hDSRTP/CZDMW (6.2) Figure 6.6 illustrates the dimensionless time required to reduce M4 to .l in cubes of precooked beef as a function of H1 Clearly, the drying time and, therefore, the drying rate are independent of H. when it is greater than approximately 100. For values of H 95 .moom pmxooompm wo monoo ow mumm mewmuanoummum ofluonamoeu< co muflm mHaEmm mo uoowmm pouowooum .m.o ouowflm musom .u 1 mega .ON .00 .Om .o¢ .om .om .OH .0 p n n P p b b monm Noum moflm N o pHHOm mum-8w\o muew m.H u z .ucoucoo ououmwoz HmwuwcH ocuoomnEo\Hmo Hooo. u x .zuw>wuusocou HmEuoLH mNn. u 0 Nu .ucmumcoo Hmpouoouum Eumuomm- Eo\sw mmoo. u L ..wooo nommcmuu mmme mommuom N Eum mm. u m .ousmmoum Emumzm ooo.mu u we .muoumuanmH ufi< o.H w ‘nuaquoo aannsiow ueaw ssaIuoisuamrq .wcquoumNmoum owpozamoeu< pops: : mo coHuocsm 6 mm mmom mo monso omxoooopm CH H. u 2 OD pommmHm oEHH mmmacowmmuewo .o.o ouswwm «8 mg; n m 96 .ooN D .ooH b .oN - .OH b .m .N .OH .ON 00m .00 .0m 9 =3 am}; ssaIuoisuamia - Zs/qq = I 97 greater than 100 the rate of mass transfer is effectively con- trolled by the internal mass transfer mechanism. The maximum possible rate of atmOSpheric freeze-drying as predicted by the approximate three-dimensional model for one centi- meter cubes of precooked beef is illustrated in Figure 6.7. A value of hD = .040 gm/cmz-sec-atm was used to compute the curve in Figure 6.7. This value of hD corresponded to HI= 148. The curve illustrated in Figure 6.7 can be used to predict the max- imum rate of atmOSpheric freeze-drying for cubes of precooked beef of any size. The ratio of drying times for two cubes of different size is proportional to the ratio of the sample size squared. Economic analysis of the atmOSpheric freeze-drying process is beyond the scope of this research, however, some observations can be made from the results presented concerning its practical usefulness. Obviously the rate of atmospheric freeze-drying is slow even in relatively small samples. The most promising area of the operating variable space is where hD is high and sample size is small. This vicinity was of interest to Malecki, gt 31. (1969) but other problems concerning fluidization of frozen particles in the fluidized bed hampered the investigation. Perhaps other configurations of equipment which could investigate this domain of the operating variable space would meet with more success. Economic viability of the process must be based on low capital investment for equipment and a continuous process. In both of these areas great improvement is possible over conventional freeze-drying. 98 .momm omxoooopm mo mmHAEwm Hmownsu mo wcfixpauoummum owumzamoeu< pom o>uso wcwzun commisseflxmz wouowoopm .m.o ouowwm muoom .u s mEHH .0m .05 .oo .om .oq .om .om .OH .0 W. n p b1 b P p a r EumuommuNEo\Ew u a: v oHHOm muo18w\onaEw m.H n 02 .ucmucoo ousumwoz HmHuHGH octoomu5o\Hmo Hooo. n M .zuHPfiuoSpcoo HmEuoLH mmm. n o .ucmuwooo Honouoouum Eu m. u m .mmocxowsHumem mHaEmm Sum no. u m .musmmmum Ewummm ooo.mu u we .muoumquESH ufi< w ‘quanuoo alnqsiow ueaw ssaluoisuamiq CHAPTER‘VII CONCLUSIONS 1. The rate of atmOSpheric freeze-drying in precooked beef was found to be adequately predicted by numerical solution of a mathematical model of simultaneous heat and mass transfer. The model assumed water vapor diffusion was the mechanism of mass transfer and thermal conduction was the heat transfer mechanism through the porous zone. 2. Nonlinear estimation of tranSport parameters in a mathematical model with an integrated dependent variable (mean moisture content) was successfully demonstrated. 3. The internal heat and mass tranSport parameters of atmospheric freeze-drying were evaluated by nonlinear estimation. The structural constant of the porous zone was evaluated at .81 for diffusion parallel to the fibers of the mean and .62 for dif- fusion perpendicular to the fibers. The mean value of the effective thermal conductivity was found to be .0001 cal/cm-sec-OC. 4. The mechanism of thermal tranSport in the porous zone of a product under freeze-dehydration was shown to be largely limited to conduction through the solid fraction of the matrix. Counter-flow of water vapor substantially reduced the contribution by the porous fraction to the total heat transfer. 99 100 5. The process of atmOSpheric freeze-drying was analyzed in cubes of precooked beef by use of an approximate three-dimensional ‘model. The operating variables of air temperature, system pressure, sample size and surface mass transfer coefficient were investigated. The rate of atmOSpheric freeze-drying was found to be strongly and directly related to air temperature. Sample size and the surface mass transfer coefficient were found to be the most promising variables to yield a practical process. Time to remove 90 percent of product moisture from one-centimeter cubes of cooked beef was approximately 30 hours for optimum conditions at atmospheric pres- sure. The drying time for other size cubes was proportional to the ratio of the sample size squared. BIBLIOGRAPHY BIBLIOGRAPHY Beck, J. V. 1966. Transient Determination of Thermal PrOperties. Nuclear Engipeeripg and Design 3: 373-381. Beck, J. V. 1969A. Class Notes from M. E. 818, Estimation in Heat and Mass Transfer, Spring Term, 1969, Michigan State University, East Lansing. Beck, J. V. 1969B. Personal Communication. Burke, R. F. and R. V. Decareau. 1964. Recent Advances in Freeze- Drying of Food Products. Advances ip Food Research 13: 1-89. Carmen, P. C. 1956. Flow pprases Through Porous Media. Academic Press, New York. Crank, J. and P. Nicolson. 1947. A Practical Method for Numerical Evaluation of Solutions of Partial Differential Equations of the Heat Conduction Type. Proceedings pf Cambridge Phil. Soc. 43 (17): 50-67. Deutsch, R. 1965. Estimation Theory. Prentice-Hall, Englewood Cliffs, N. J. Draper, N. R. and H. Smith. 1966. Applied Regression Analysis. John Wiley and Sons, Inc., New York. Dunoyer, J. M. and J. Larousse. 1961. Experiences Nouvelles sur la Lyophilisation. Trans. pf Eighth Vacuum Symposium and Second International Congress 2: 1059-1063. Dyer, D. F. and J. E. Sunderland. 1966. Bulk and Diffusional TranSport in the Region Between the Molecular and Viscous Conditions. International Journal pf Heat and Mass Transfer 9: 519-526. Dyer, D. F. and J. E. Sunderland. 1968. Transfer Mechanisms in Sublimation Dehydration. Journal pf_Heat Transfer, ASME Trans. 90C: 379-384. Eckert, E. R. G. and R. M. Drake, Jr. 1959. Heat and Mass Transfer. McGraw-Hill Book Co., Inc., New York. 101 102 Evans, R. B., G. M, Watson, and E. A. MaSOn. 1961. Gaseous Diffusion in Porous Media at Uniform Pressure. Journal gfi Chemical Physics 35: 2076-2083. Gunn, R. D. 1967. Mass Transport in Porous Media as Applied to Freeze-Drying. Ph.D. Dissertation, University of California, Berkeley. Gunn, R. D. and C. J. King. 1969. Mass Transport in Porous Materials Under Combined Gradients of Composition and Pressure. A.I.Ch.E. Journal 15(4): 507-514. Harper, J. C. and A. L. Tappel. 1957. Freeze-Drying of Food Products. Advances ip_Food Research 7: 172-235. Harper, J. C. 1962. TranSport Properties of Cases in Porous Media at Reduced Pressures with Reference to Freeze- Drying. A.I.Ch.E. Journal 8(3): 298-302. Harper, J. C. and A. F. El Sahrigi. 1964. Thermal Conductivities of Gas-Filled Porous Solids. Industrial and Engineering Chemistry, Fundamentals 3(4): 318-324. Hohner, G. A. and D. R. Heldman. 1970. Computer Simulation of Freezing Rates in Foodstuffs. A paper to be presented before the Institute of Food Technology, San Francisco, Calif., May, 1970. Kan, B. and F. deWinter. 1966. The Acceleration of the Freeze- Drying Process Through Improved Heat Transfer. Paper presented at the 26th Annual Meeting of the Institute of Food Technology, Portland, Oregon, May, 1966. Krischer, O. 1959. Die Wissenschaftlichen Grundlagen der Trockungstechnik. Springer, Berlin. Lambert, J. B. and W. R. Marshall, Jr. 1961. Heat and Mass Transfer in Freeze-Drying. In: Freeze-Dgying pf Foods., NAS-NRC publication, Frank R. Fisher, editor. 105-133. Lentz, C. P. 1961. Thermal Conductivity of Meats, Fats, Geletine Gels and Ice. Food Technology 15: 243-247. Lewin, L. M. and R. F. Mateles. 1962. Freeze-Drying Without Vacuum, a Preliminary Investigation. Food Technology 16(1): 94-97. Malecki, G. J., P. Shinde, A. I. Morgan, Jr., and D. F. Farkas. 1969. AtmOSpheric Fluidized-Bed Freeze-Drying of Apple Juice and Other Liquid Foods. Paper presented at the 29th Annual Meeting of the Institute of Food Technology, May, 1969, Chicago, Ill. 7h 103 Meeter, D. A. 1964. Program GAUSHAUS, Numerical Analysis Lab- oratory, University of Wisconsin, Madison. Meryman, H. T. 1959. Sublimation Freeze-Drying Without Vacuum. Science 130: 628. Meryman, H. T. 1964. Induction and Dielectric Heating for Freeze- Drying. In: ASpects Theoriques pg Industriels d2 13 Lyophilisation. 65-68. Mink, W. H. and G. F. Sachsel. 1961. Evaluation of Freeze-Drying Mechanisms Using Mathematical Models. In: Freeze-Dryipg ug£_Foods, NAS-NRC publication, Frank R. Fisher, editor. 84-920 Ngoddy, P. O. 1969. A Generalized Theory of Sorption Phenomena in Biological Materials. Ph.D. Dissertation, Michigan State University, East Lansing. Peng, K. C. 1967. The Desigp and Analysis pf Scientific Experi- ments. Addison-Wesley Co., Reading, Mass. Perry, J. H. 1950. Chemical Engineers' Handbook. McGraw-Hill Book Co., Inc., New York, 3rd Ed. Pfahl, R. C., Jr. and B. J. Mitchell. 1969. A General Method for Simultaneous Measurement of Thermal Properties. Pre- print of a paper for presentation before the American Inst. of Aeronautics and Astronautics. Riedel, L. 1957. Kalorimetrische Untersuchen fiber das Cefrieren von Fleisch. Kfiltetechnik 9(2): 38. Sandall, O. C. 1966. Interactions Between Heat and Mass Transfer in Freeze-Drying. Ph.D. Dissertation, University of California, Berkeley. Sandall, O. C., C. J. King and C. R. Wilke. 1967. The Relation- ship Between Transport PrOperties and Rates of Freeze- Drying of Poultry Meat. A.I.Ch.E. Journal 13(3): 428-438. Saravacos, G. D. and M. M. Pilsworth, Jr. 1965. Thermal Con- ductivity of Freeze-Dried Model Food Gels. Journal pf Food Science 30: 773-776. Saravacos, G. D. and R. M. Stinchfield. 1965. Effect of Temper- ature and Pressure on the Sorption of Water Vapor by Freeze- Dried Food Materials. Journal of Food Science 30(5): 779-785. Scheidegger, A. E. 1957. The Physics p£_Flow Through Porous Media. MacMillian Co., New York. 104 Scott, D. S. and F. A. L. Dullien. 1962. Diffusion of Ideal Cases in Capillaries and Porous Solids. A.I.Ch.E. Journal 8: 113-117. Short, B. E. and H. E. Staph. 1951. The Energy Content of Foods. Ice and Refrigeration 121(5): 23. Smith, G. D. 1965. Numerical Solution pf Partial Differential Equations. Oxford University Press. New York and London. Snedecor, G. W. 1956. Statistical Methods. Iowa State University Press, Ames, Iowa. Threlkeld, J. L. 1962. Thermal Environmental Engineering. Prentice- Hall, Inc., Englewood Cliffs, N. J. Triebes, T. A. and C. J. King. 1966. Factors Influencing the Rate of Heat Conduction in Freeze-Drying. Ind. and Eng. Chem., Process Design and Deve10pment 5(4): 430-436. Wakao, N., S. Otani, and J. M. Smith. 1965. Significance of Pressure Gradients in Porous Materials: Part 1. Diffusion and Flow in Fine Capillaries, A.I.Ch.E. Journal 11: 435-439. Woodward, H. T. 1961. Study of Vapor Removal Systems in Dehydration of Food Products Having Piece or Block Conformation. Quartermaster Contract Report (DA 19-129-QM-1597). Woodward, H. T. 1963. Freeze-Drying Without Vacuum. Food Engineering 35(6): 96-98. APPENDICES APPENDIX I DERIVATION OF THE ENERGY EQUATION, EQUATION (3.1) From an energy balance at constant pressure on a differ- ential volume of the porous zone, dV = Adx, the following equation for transport in the (x) direction is obtained: dem +MC )aLA 1%: - ‘91 +c (at) -c (m) +deAHvbfl. as 107 APPENDIX III DERIVATION OF FINITE DIFFERENCE APPROXIMATIONS OF THE HEAT AND MASS TRANSFER EQUATIONS, EQUATIONS (3.18) AND (3.19) Finite difference approximations of the heat and mass trans- fer equation were required for the numerical solution. Considering the heat transfer equation first and using the backward-difference approximation (Smith, 1965), the change in internal energy in differential volume element dV = Add during time step A9 can be written + +1 + Tn 1_Tn Tn+l_Tn+l Tn -T? 1 A¢p(c +MC )n ..1___.1. = k 1+1 1 _ 1 1- n+1 n+1 n+1 n+1 n+1 (A.10) De 1+1- 1"]. 1+1- i-l Mi ..Mi + —— + —— . CPWD 2M) A 2 ACDPAHV A9 Equation (A.10) is rearranged to yield equation (3.19), n n+1 n n+1 n+1 n+1 + _ = - 9(0pd Mpr)i(Ti Ti) ZZ(Ti+1+Ti_1 2Ti ) (3.19) C 2 pw n+1 _ n+1 n+1 n+1 n+1 n + 4 (Pi+l Pi-l)(Ti+l Ti-l)+pAHv(Mi Mi)° The mass transfer equation in finite difference from is derived from a mass balance on the void Space of a differential volume element, edV = eAd¢, during time step A9. The Crank- Nicolson approximation (Smith, 1965) is used and all coefficients are evaluated in the (n)th time frame. 108 109 n+1 n :1. = A9 + + + D [Pu l n l Pn _Pn Pn+1 _ Pn l Pn _ Pn :} M n A$._Jl.t.JL_.SM. RT P dr . sat 1 (A.1l) _gi-1'Pi +i-1 i_i i+1_i 1+1 2D 495 M5 M M Equation (A.Il) is rearranged to yield equation (3.18), M n w p dM n+1 n Z n+1 n+1 n+1 n n n —— + - = — - + -2 . . . CRT Psat dr)i(Pi Pi) 2Ei+1+Pi-1 2Pi +Pi+1 Pi-1 Pi] (3 18) APPENDIX IV COMPUTER PROGRAM MAIN AND SUBROUTINE MODEL PPOGRAM MAIN DIMENSION W(100)6TIM(100) DIMENSION THI3IQSIGNS(3)QOIFF(3I COMMON TSQIIMQDAWQMIQSSTWHQPS EXTERNAL MODEL NP =3 DOIPJ=IOI RFADIfiOo?) NORqNTFST 7 FORMAT(?IS) DO 10 I=19NOH 10 RFAD(6099) WIIIoTIMII) 9 FORMAT(F8.10F10.II PFADU‘JOvB) IQQUAWQMIQCH IWRQPSQIH( I.)9IH(2) OIHIRI 8 FORMATI9F8.0) DO 11 I=IvNP SIGNS(I)=1. ll DIFF(I)=.01 EpSl=IOOE"3 EPS2=I.0E-? CALL GAUSHAUS(NTFST.MHHEL.NOH.w.NP.TH.DIFF.SIONS.EPSlo leSZOZOQOOIQIOO) 12 CONTINUE END 110 111 SUSPOUTINE MODEL (NPHHHoTHQWFToNUHoNP) DIMENSION THI1)oWFTIl)-TIM(100) DIMENSION T(31)6PT3I)oMTRI).AT30.31)9AA(31)oAS(31) DIMENSION OIDM(3l)oPSAT(3I)0DMDP(SIIoPI(3])9T1I31) DIMENSION OLOPT31).TST3I) COMMON TSOTIMQUAWQMIQSOTWHOpS RFAL MoMIoMF C CONSTANTS OF THE MODEL ARE LISTED HFLflw C UNITS OF THE MODEL ARF CM SEC CM ATM CALORIE K OCDF)O(1C)OCDC)OCOF)OCDC) O C(DC30 RS: BULK DFNSITY OF DHY PPOOUCT CPW= SPECIFIC HEAT OF WATER CPS = SPECIFIC HEAT OF OPY PROOUCT E = POROSIIY OF OHY PPOOUCT DAW = FREE GAS MUTUAL DIFFUSIVITY. AIH AND WATER VAPOR PA = VAPOR PRESSURE OF AIR STREAM DHV = HEAT OF VAPOHI/ATIUN OHS = HEAT OF SURLIMAIION. ICF MI= INITIAL MOISTUPF CHNTFNT S = HALF THICKNFSS OF SAMPLE NN= NUMHFR OF SPACF NOOFS TIMWH = TIMF REOUIHFO IO ”HOP CENTEHLINE TFMP TO MET- HULH TFMPFHAIUPt OF AIR STREAM THR = WET-BULB TEMPPHAIURF Of AIR STRtAM MF SATURATION EOUILIHRIUM MDISTHPF CONTFNI PS SATURATION VAPOR PPFSSUPE OF ICE AT TWH FK TH(3) RS3046 pr=lo CPS=.38 CPC= 1.19 =.76 pA=O .0 DHV=HSO. DHS= 676. NN=11 TIMWB= 300. Mon? MODEL ASSUMES SUHLIMAIION 18 FROM FREE ICE SURFACE wHILE CENTEPLINE DROPS FROM I-AIH TO I-WFI HHLH ENN=NN KK=2 HFTII) 31.0 DF=TIMWR*IH(I)“(PS-PA)*S/(HS*(MI-MF)*DAW) F=l.'DF DR=1./(ENN-IOI DT=206*DR*DAw/(S*IH(III K=0 DO 10 I=19NN TIII=TWR pIII=PS 10 MIII=MI ll 12 20 21 112 ~ TIME =TIMNH-OT NNl=NN-l NFHOLO=NN DPHLD=DF Z7=TH(3)*DT/((DR**2)*nAw) H=DT*TH(1)*S/(DAW*DH) TC=TWB K=K+l TIME=TIMESDT IF(F.LE.O.) F=0. IFIK.GT.1) NFHDLD=NF NF=(10’FI/DR NF=NN-NF IF(FOEQOOO) NFz? IFINFHDLD.LF.NF) GO TO [2 RATIO=((ENF-2.)*HH-F)/((FNF-l.)*DH-F) T(NF)=T(NF?)+RATID*(T(NF+II-TINFBI) PINFI=PINF2)+RATIUFIPTNFtII-P(NF2)I MTNF)=M(NF?)+RAIIO*(MTNF+I)-M(NF?)I XX=TC-243.IS P(NF-l)=(.ZSSRHAAOAJ76+.044891607OH09xx-.00051744a92312 l *(XX**2)+.0001?993D?IH95*(XX**3II/760. M(NF-1)=MF T(NF-l)=TC IF(K.GT.II ORHannu? NF2=NF-l NF1=NF+1 ENF=NF DQ2=I.~F-(ENN-ENFI*HH DPZHR=(OR8+ORHLO I/P. IF(NFHOLD.GT.NF) OPPHPzOH) DO 20 I=NF?.NN OLOPIII=PTII 0LOM(I)=M(I) xx=T(I)-243.IS PSAT(II=(6259886464376+.O44R9lb0IOH0*XX-.00061794892312* l *(XX”*?)+.0001?9930?189S*(XX**3II/760. DMOP(I)= .R/PSATII) P1(I)=.219*F/T(I) +RS*UMDP(I) IF(F.GT.0.I OLOPINFPI=PSAI(NTR) Z=OT*TH(2)*.43H/((IIMFP>+T(NN))*(OH**2)I IFIFOGTOOO) ”LIIM(“IF?)=:IF IF(1.-F-OR) 21.71.22 T(NN)=TS TTNF2)=TC P(NF2)=PSAT(NF?I PITNN)=2.*PI(NN) P?=P1(NN)+Z*((DH/OR?)**2)+S*TH(1)*UT/( 0R2) P(NN)=(PI(NNI*OLOP(NF)+7*((DR/UH?)**P)*IP(NFZI+OLOP(NF2) -OLDP(NF))-TH(1)“DT*IDLDPINFI-2.*PA)/(DR2)I/PZ DP=(P(NF)-P(NF?)I/OPP OLDDP=(OLDP(NF)-OLOP(MF?)IIIOPHLOI F=F*.S*S*l*(DR**2)*(HP+0LODD)/(RS*(MI-MF)) TC=TC+(DHS*Z*.S*(OP+OLUOPI+Z7*(I(NFI-TTNFEI)/OH2)*(OR**2) 1 19 22 23 24 26 27 I 28 25 30 31 1 35 40 SD 49 l 113 *S/ (Fwsfl I . {M'Ifiwm MINFZI=MF MINNI= .8*P(NN)/PSAT(MN) WT=F+.S*(l.-FI*(MF+~(NN)I/MI GO TO 101 IFIF) 23923074 AI].1)=P1(1)+ Z A(192)=- Z AIl.NN+l)=(P1(l)- 7)*P(I)+ 2*PTP) GO TO 25 IFTORZ-l.0F-SI 26.26.27 AINFQNF-I) =0.0 AINFoNFI=I.0 A(NF.NF+1) =0.0 A(NF¢NN+1I=OLDP(MF) GO TO 28 A(NFqNF-1)= -Z*OH/Dv? A(NFQNF+I)='Z A(NF9NF)=2.*P1(NF)*(.§+HR?HR/DRI-A(NFoNFZI-AINFQNFI) A(NF,NN+1)=P(NF)*(A(NF.NF)+A(NF.NF-II+A(NF9NF+1)) -A(NF.NF-I)*(P(NF-II-PINF)I+A(NF.NF+1)*(P(NFI-P(NF+1II AINF2.NF2)=1.0 AINF2.NF)=0.0 ATNonNN+II=PSATINFPI IF(NF.EQ.NN1) GO TO 31 II=NF IFIF.GT.0.) II=NFI DD 30 I:IIONN1 AIIQI'I)=’7 A(I.I)=?.*(PI(I)+/) AII.I+1)=-Z AIIoNN+II= ZSPTI-I)+a.*(91(II-Z)*P(I)+Z*P(I+l) A(NN.NN-1)= .25*Pl(HN)-/ A(NN.NN)= .75”PI(MNI+/+H A(NN.NN»1)=( .?S*QI(MM)+/)*P(NN-II+(.7S*P1(NNI-Z-HI sPTNN)+2.*H*PA ASTNFZ )=A(NF-I.NM+I) AAINF2 )=A(NF-l.NF-I) OO 3SI=NFQNN AATTI=ATIcI)-A(I.I-I)*A(I-l.I)/AA(I-lI AS(I)=A(I.NN+1)-A(I.I-I)*AS(I-lI/AA(l-l) PTNN)=AS(NN)/AA(NN) NNF=NN-NF2 on 40 I=19NNF J=NN'I pIJI=IAS(J)-A(J.J+I)*P(I+I ))/AA(J) IFIF.GT.0.) DINFZI=PSAIINF?) DD 50 I=NFP~NN IMIII= .?*P(T)/PSAT(I) IFIF.GT.0.) MINFPI=MF OLOOP=DP IFIDR2-1.0F-3) 48.48.49 DP=IDR2+DR)*(PINF)-p(NF?)I/(DH*OH?) +DR?* (P(NF2) -P(NF1))/(DR*(OR+OH?)) 48 47 58 59 60 SI 54 SS 96 S3 70 71 80 114 GO TO 47 DP=IPINFII'P(NF2II/(HR+HRPI IFIIIIS‘TINF2II/ISI.hT. .001) I“) H) 59 TC=TS DO Q8 I=NF?0NN TIII=TS GO TO 01 DO 60 I=NF20NN TIII)=RS*(CPS+M(II*CPM) TQII)=RS*UHV”IM(II-HI“MIIII IFIFofiT. .001) G” T“ 99 AIIQII=II (I)‘7l'0351?/*CI"U*IP(1)-P(?)+(ILIIPIII’ULUP(?)I AI192)= .?S*7*CPW*IPIII‘P(2)+HLHP(II’ULHP(2)I 'Zl AIIONN*I)=T(II*(II(I)*8.'A(IQIII’T(?)*A(Ic?)+l5(1) GO TO 53 IFIDR? 'IoOE'S) 54-54-55 AINFZ’NFE) =10 A(NF20NFI=000 A(NF?9NN*II=TC AINFQNle 30.0 AINFQNF)=I.D AINFQNFI) 30.0 A(NFQNN*I) =T(NF) GO TO 53 AINFZoNFKI=F*QS*CPC*(I.+MII* ll“S*(DR**2I/UR2 A(NF?9NF)=- S*//*(HH**/I/HH? AINFZQNN+II=T(NF2)*I*US*(I.+MI)*C“C+ DHS*S*I*(DH**?I*HP T2: (.54HR2Hw/“HI*II(NFI T1=ZZ T4= 71*Up/DR? T6=-.25*7*CPN*(PINFII+P(NF)-?.*P(MF?))/(.Q+Dw//HH) ATNFoNF2)=-T4-?.*Ih ATNFoNFI=T2+T3+T4+IA A(NF9NF+1)=-T3+T6 AINF.NN+I)=T(NF)*T? II=NF ' IFTF.GT.0.) II=MFI IFIII.GT.NNI) UO TO 71 DO 70 I=II-NNI AIIoI-I)=-ZZ+.ZH*CPW*/*(P(I+I)-P(I-II) AIIoII=T1(T)+77*P. AII9I+1I=-A(IqI-l)-7/*R. AIIoNN+II=TI(I)*T(I) IF(F.LF. 0.0) A(I~NN+I)=TI(I)*T(I)+Is(II CONTINUE A(NN9NN-I)=0.0 ATNN;NN)=I. ATNNoNN+II=TS AS(NF2)=A(NF?.NN+1) AATNFPI=ATNF26NFPI DO 90 I=NFQNN AAII)=A(Tol)-A(I-I-I)fiAII-l-II/AAII-l) AS(I)=A(I0NN+l)-A(IoI-II*AS(I'II/AAII'II T(NN)=AS(NN)/AA(NN) 90 91 100 101 122 102 103 106 107 104 115 0090 1:1.NNF JzNN-I TIJ)=(AS(J)-A(J~J+lI4‘II J+l))/AA(.II WT=F§MI+.‘S*DR?*(M(NF?I+MINFII IFIF.EQ.0.0) WT=.S“DH*(M(I)+M(?)) DO 100 I=NF9NNI WT=WT+.S*DP*(M(I)+M(I+I)) WT=WT/MI IFIARSF(TIMf-TIM(KKII.OT. .8901) 60 I0 102 WFT(KK)=WI WQITEIOIqIPE) WFTTKKI FORMAT(* *oF6.4) KK=KK+I IF(F.E0. 0.0 .OR. NF.FO. NN) GO TO 103 DFP=Z* (I)R**?)*.S* (I)P+()LDDT—’)*S/ (RS* (M1-MF)) F=F+DFP IF(KK.GT.NOR) GO TO 104 IF(M(1).GT. .02) GO TO 11 DO 107 I=KK9NOH HFTII) =0.0 RETURN END APPENDIX V SUMMARY OF RESULTS FROM PARAMETER ESTIMATION TESTS Test number: 1 Fiber orientation: parallel 116. Air Temperature: -8.20C Sample Half-thickness: .477 cm System Pressure: .97 atm Initial Moisture Content: 1.477 Dimensionlesz Time Experimental Dimensionless Computed 9x10" Mean Moisture Content, M Residual 0.00 1.000 .000 .38 .959 .008 1.70 .876 .002 2.30 .847 .000 2.69 .833 .001 3.07 .815 -.003 6.62 .704 .008 7.29 .679 .002 8.05 .657 .000 8.92 .635 —.001 9.21 .628 -.001 9.79 .614 -.001 10.55 .600 .002 11.13 .585 .000 11.51 .576 -.001 11.90 .569 .000 15.54 .495 -.004 16.12 .485 -.003 16.88 .474_ -.001 17.65 .465 .003 18.04 .456 .000 18.61 .442 -.004 19.00 .436 -.004 19.86 .416 -.010 20.43 .409 -.008 21.11 .402 -.005 25.71 .337 -.003 26.29 .330 -.002 26.86 .327 .003 27.63 .319 .005 28.40 .310 .006 30.70 .280 .006 36.36 .215 .009 117 Test number: 2 0 Fiber orientation: parallel Air temperature: -8.2 C Sample half-thickness: .420 cm System pressure: .97 atm Initial moisture content: 1.505 Dimensionless Time Experimental Dimensionlegs Computed 9x10'4 Mean Moisture Content, M Residual .00 1.000 .000 .43 .955 .002 .87 .924 -.003 1.30 .899 -.003 1.74 .877 .002 2.17 .854 .001 2.61 .837 .004 7.06 .658 .001 7.82 .635 .002 8.48 .613 .000 9.13 .594 .000 9.56 .585 .003 10.36 .561 .000 11.08 .543 .000 11.95 .517 -.004 12.61 .500 -.005 13.26 .490 .000 17.39 .408 .005 18.26 .389 .003 19.13 .372 .002 20.00 .354 .000 20.86 .337 -.001 21.30 .330 .000 22.39 .309 -.002 23.04 .299 -.001 23.69 .285 -.004 28.26 .208 -.008 29.12 .191 -.012 30.00 .181 -.009 30.86 .167 -.010 31.73 .153 -.012 32.82 .139 -.010 33.47 .129 -.011 34.12 .118 -.013 38.25 .083 .007 39.12 .070 .011 39.99 .063 .025 118 Test number: 3 0 Fiber orientation: parallel Air temperature: -2.8 C Sample half-thickness: .405 cm System pressure: .97 atm Initial moisture content: 1.723 Dimensionless Time Experimental Dimensionlegs Computed 6x10" Mean Moisture Content,,M Residual .00 1.000 .000 .48 .940 .015 1.09 .883 .010 1.58 .843 .002 2.06 .800 -.014 2.55 .760 -.027 2.91 .733 -.029 3.40 .700 -.032 7.64 .526 -.016 8.12 .513 -.012 8.61 .496 -.012 9.34 .472 -.012 10.06 .451 -.011 10.55 .436 -.011 11.16 .419 -.010 11.52 .409 -.009 13.10 .366 -.008 14.67 .319 -.015 19.52 .215 -.005 20.01 .205 -.006 20.49 .194 -.006 21.10 .184 -.003 21.83 .172 -.001 22.80 .159 .006 23.28 .152 .009 119 Test number: .4 0 Fiber orientation: perpendicular Air temperature: -2.8 C Sample half-thickness: .472 cm System pressure: .97 atm Initial moisture content: 2.01 Dimensionless Time Experimental Dimensionlegs Computed 9X10-4 Mean Moisture Content, M Residual .00 1.000 .000 .43 .937 .003 .86 .893 -.002 1.29 .857 -.001 1.72 .825 -.004 2.15 .799 -.007 2.58 .773 -.004 3.43 .731 .003 3.86 .708 .002 4.29 .689 .004 4.72 .671 .006 5.15 .653 .006 9.01 .515 .006 9.66 .498 .009 10.30 .476 .005 10.95 .452 -.000 11.66 .439 .006 12.23 .420 .003 12.77 .407 .004 13.88 .378 .003 14.30 .368 .003 14.74 .355 .000 15.17 .345 .000 19.32 .248 -.005 19.96 .235 -.004 20.61 .222 -.004 120 Fiber orientation: perpendicular Sample half-thickness: .460 cm Initial moisture content: 1.688 Test number: 5 0 Air Temperature: -2.8 C System pressure: .92 atm Dimensionless Time Experimental Dimensionlegs Computed exlo' Mean Moisture Content, M Residual .00 1.000 .000 .38 .943 .001 .75 .911 .000 1.13 .881 .003 1.50 .854 .002 1.88 .830 .000 2.73 .782 -.004 3.29 .760 -.001 3.85 .734 -.005 7.57 .602 .002 8.06 .585 .001 8.64 .572 .007 9.21 .554 .006 9.78 .539 .007 10.34 .525 .010 10.81 .511 .009 11.84 .489 .014 12.22 .477 .012 12.60 .469 .014 16.54 .359 -.004 17.10 .346 -.005 17.67 .336 -.003 18.23 .311 -.005 19.36 .299 -.006 19.93 .288 -.006 20.87 .271 -.004 21.25 .265 -.005 21.53 .260 -.003 21.90 .254 -.002 25.47 .196 .004 26.04 .188 .006 121 Test number: 6 0 Air temperature: -2.8 C System pressure: .97 atm Fiber orientation: perpendicular Sample half-thickness: .405 cm Initial moisture content: 1.757 Dimensionless Time Experimental Dimensionlegs Computed exlO'4 Mean Moisture Content, M Residual .00 1.000 .000 .48 .932 .001 1.13 .869 -.011 1.45 .846 -.013 1.94 .816 -.016 2.43 .790 -.023 3.15 .756 -.018 3.64 .734 -.016 4.17 .711 -.017 5.34 .669 -.013 5.82 .653 -.012 6.31 .634 -.015 6.79 .620 -.014 11.64 .490 -.012 12.37 .476 -.009 13.10 .459 -.010 13.82 .443 -.010 14.60 .424 -.012 15.28 .411 -.011 16.81 .380 -.012 17.46 .367 -.012 18.19 .353 -.013 23.52 .262 -.010 24.73 .245 -.008 25.46 .233 -.008 26.19 .222 -.008 26.92 .213 -.005 27.40 .206 -.005 34.00 .133 .016 122 Test number: 7 Fiber orientation: perpendicular Air temperature: -8.20C Sample half-thickness: .477 cm System pressure: .97 atm Initial moisture content: 1.813 Dimensionlesz Time Experimental Dimensionlegs Computed exlo‘ Mean Moisture Content, M Residual .00 1.000 .000 .46 .956 .004 .92 .921 -.003 1.39 .898 .001 2.36 .844 -.001 2.82 .824 -.001 3.28 .805 -.002 3.74 .791 .002 8.36 .662 .006 9.05 .646 .006 9.79 .626 .003 10.16 .616 .001 10.86 .597 -.003 11.55 .582 -.004 12.01 .572 -.005 13.28 .549 -.004 13.97 .536 -.005 14.67 .522 -.006 18.94 .459 .001 19.55 .449 .001 20.33 .438 .001 21.02 .429 .003 21.71 .420 .004 22.17 .410 .000 22.87 .399 -.001 24.03 .382 -.002 24.71 .374 -.001 25.41 .363 -.002 29.80 .308 -.001 30.49 .300 -.001 31.18 .292 .000 31.87 .283 -.001 32.57 .276 .000 33.26 .269 .001 33.95 .262 .002 35.10 .251 .004 35.80 .242 .003 36.49 .236 .005 40.61 .182 .005 123 Test number: 8 0 Fiber orientation: perpendicular Air temperature: -8.2 C Sample half-thickness: .472 cm System pressure: .97 atm Initial moisture content: 2.01 Dimensionlesz Time Experimental Dimensionless Computed 9x10' Mean Moisture Content, M Residual .00 1.000 .000 .36 .946 -.006 .90 .894 -.024 1.26 .872 -.023 1.80 .843 -.021 2.33 .816 -.023 2.69 .799 -.025 3.41 .772 -.025 3.95 .749 -.025 4.49 .732 -.022 8.26 .619 -.021 8.88 .606 -.020 9.34 .592 -.021 9.88 .581 -.019 10.41 .568 -.020 10.90 .556 -.021 12.43 .525 -.019 12.93 .515 -.019 13.47 .505 -.018 17.42 .438 -.012 18.38 .425 -.009 18.86 .417 -.009 19.40 .410 -.007 19.93 .401 -.008 20.48 .392 -.008 21.02 .388 -.003 25.33 .325 -.002 25.87 .321 .001 26.76 .307 .000 27.30 .301 .001 29.10 .278 .002 29.64 .273 .004 124 Test number: 9 0 Fiber orientation: parallel Air temperautre: -2.8 C Sample half-thickness: .385 cm System pressure: .58 atm Initial moisture content: 1.488 Dimensionless Time Experimental Dimensionlegs Computed 9x10"4 Mean Moisture Content, M Residual .00 1.000 .000 .45 .957 .013 .90 .924 .015 1.80 .863 .017 2.70 .809 -.008 3.59 .766 .004 4.49 .722 .006 5.39 .686 .010 6.29 .647 .007 8.54 .567 .005 9.44 .535 .002 10.33 .513 .007 11.23 .488 .008 19.77 .283 .004 20.67 .261 .000 21.57 .243 .001 125 Test number: 10 0 Fiber orientation: parallel Air temperature: -2.8 C Sample half-thickness: .394 cm System pressure: .58 atm Initial moisture content: 1.48 Dimensionless Time Experimental Dimensionless Computed 9x10' Mean Moisture Content, M Residual .00 1.000 .000 .86 .914 .014 1.72 .849 .013 2.57 .795 -.021 4.81 .671 .003 5.58 .637 .008 6.43 .599 .011 7.29 .562 .010 8.15 .531 .015 15.87 .267 -.002 16.73 .243 -.002 17.59 .216 -.007 18.44 .192 -.009 19.30 .178 -.002 20.16 .161 .002 126 Test number: 11 Fiber orientation: perpendicular Air temperature: -2.8OC Sample half-thickness: .450 cm System pressure: .58 atm Initial moisture content: 1.027 Dimensionlesz Time Experimental Dimensionless Computed 9x10- Mean Moisture Content, M Residual .00 1.000 .000 .33 .922 -.003 .66 .901 .013 1.32 .845 .008 1.97 .797 .001 2.30 .780 .001 3.95 .698 -.005 4.60 .672 -.007 5.26 .642 -.014 5.92 .620 -.018 12.41 .439 .002 13.55 .417 -.001 14.14 .396 -.002 15.13 .370 -.007 16.12 .344 -.004 17.10 .318 -.008 18.09 .296 -.009 19.89 .258 -.010 20.88 .236 -.012 21.87 .219 -.010 27.96 .111 -.010 28.94 .096 -.009 29.92 .085 .008 30.92 .079 .028 31.24 .072 .027 127 Test number: 12 Fiber orientation: parallel Air temperature: -2.80C Sample half-thickness: .402 cm System pressure: .58 atm Initial moisture content: 1.61 Dimensionless Time Experimental Dimensionless Computed 9x10'4 Mean Moisture Content, M Residual .00 1.000 .000 .41 .956 .013 .83 .951 .005 1.65 .845 -.009 2.48 .797 -.019 3.31 .753 -.017 4.13 .720 -.007 5.79 .655 .001 6.61 .624 .001 7.44 .596 .003 8.26 .568 .002 9.09 .544 .004 16.94 .339 .001 17.76 .314 -.006 18.60 .297 -.006 19.42 .279 -.006 20.25 .262 -.007 21.07 .249 -.003 21.90 .229 -.007 128 Test number: 13 Fiber orientation: parallel Air temperature: -2.80C Sample half-thickness: .422 cm System pressure: .58 atm Initial moisture content: 1.57 Dimensionless Time Experimental Dimensionless Computed 9x10" Mean Moisture Content, M Residual .00 1.000 .000 .37 .959 .008 .75 .927 .011 1.50 .869 .013 2.25 .818 .001 3.13 .769 .007 3.75 .738 .008 4.50 .705 .012 5.25 .671 .010 5.62 .657 .012 7.74 .579 .011 8.50 .553 .009 9.24 .531 .011 10.00 .509 .011 17.25 .314 -.002 18.00 .297 -.003 18.75 .279 -.005 19.50 .263 -.006 20.62 .241 -.005 21.75 .219 -.004 23.62 .199 .010 129 Test number: 14 o Fiber orientation: perpendicular Air temperature: -2.8 C Sample half-thickness: .461 cm System pressure: .58 atm Initial moisture content: 1.62 Dimensionless Time Experimental Dimensionleps Computed 9x10'4 Mean Moisture Content, M Residual .00 1.000 .000 .47 .936 .006 2.04 .811 -.010 2.67 .775 -.014 3.30 .745 -.009 3.93 .714 -.010 10.06 .498 -.010 10.68 .481 -.010 11.37 .464 -.009 12.10 .445 -.009 12.73 .425 -.014 13.36 .410 -.013 14.14 .392 -.013 15.02 .374 -.012 17.13 .331 -.010 17.75 .317 -.008 18.38 .306 -.006 19.01 .295 -.005 25.26 .190 .005 26.14 .178 .008 27.02 .165 .010 27.90 .153 .013 28.90 .140 .016 130 Test number: 15 Fiber orientation: parallel Air temperature: -2.80C Sample half-thickness: .425 cm System pressure: .97 atm Initial moisture content: 1.575 Dimensionless Time Experimental Dimensionless Computed QxlO'4 Mean Moisture Content, M Residual .00 1.000 .000 .44 .956 .007 .89 .918 .001 1.33 .884 .004 1.78 .853 .006 6.22 .635 .025 6.67 .615 .025 7.11 .586 .017 7.60 .570 .022 8.00 .546 .022 8.44 .512 .000 8.89 .494 .000 9.33 .478 .000 10.44 .437 .000 11.11 .415 .001 11.55 .400 .001 12.00 .386 .001 12.44 .372 .002 16.67 .257 .011 17.56 .233 .011 18.00 .222 .012 18.49 .210 .012 19.29 .195 .017 20.07 .176 .017 131 Test number: 16 Air temperature: System pressure: Fiber orientation: parallel Sample half-thickness: .460 cm Initial moisture content: 1.25 -2.8°c .97 atm Dimensionless Time Experimental Dimensionless Computed QxlO'4 Mean Moisture Content, M Residual .00 1.000 .000 .38 .942 .014 .76 .897 .006 4.36 .653 -.008 4.74 .637 -.006 5.12 .617 -.007 5.50 .600 -.007 5.88 .589 -.001 6.73 .557 .002 7.11 .542 .002 7.49 .530 .004 8.38 .497 .003 9.11 .474 .005 9.48 .464 .007 9.86 .450 .006 13.66 .330 -.003 14.04 .317 -.006 14.61 .300 -.008 15.18 .284 -.010 15.65 .272 -.009 17.74 .219 -.012 132 Test number: 17 Air temperature: System pressure: Fiber orientation: parallel Sample half-thickness: .465 cm Initial moisture content: 1.479 -8.2°c .97 atm Dimensionless Time Experimental Dimensionless Computed 9x10“ Mean Moisture Content, M Residual .00 1.000 .000 .36 .958 .008 .72 .924 -.001 1.07 .898 -.003 1.43 .870 -.007 2.51 .809 -.014 2.86 .789 -.023 3.22 .773 -.019 3.49 .760 -.019 6.80 .642 -.007 7.16 .627 -.010 7.70 .611 -.009 8.23 .598 -.006 8.83 .578 -.009 9.31 .567 -.007 10.02 .549 -.006 10.74 .528 -.009 11.01 .523 -.007 11.28 .517 -.006 12.17 .497 -.005 16.23 .405 -.009 16.65 .400 -.005 17.18 .387 -.008 17.72 .377 -.007 18.26 .366 -.008 19.57 .343 -.007 20.22 .332 -.006 20.76 .322 -.006 24.16 .264 -.007 24.88 .255 -.004 25.59 .242 -.006 133 Test number: 18 0 Fiber orientation: perpendicular Air temperature: -8.2 C Sample half-thickness: .425 cm System pressure: .97 atm Initial moisture content: 1.442 Dimensionless Time Experimental Dimensionless Computed 91*:10‘4 Mean Moisture Content1 M Residual .00 1.000 .000 .43 .955 .005 .86 .920 -.005 2.71 .828 -.005 3.43 .796 -.011 8.03 .669 -.007 8.57 .657 -.007 9.21 .642 -.009 9.42 .629 -.017 10.07 .600 -.020 12.29 .572 -.018 12.86 .563 -.016 13.33 .553 -.018 13.80 .548 -.015 18.68 .478 -.007 19.28 .469 -.007 20.00 .461 -.005 20.57 .452 -.006 21.25 .443 -.005 21.85 .435 -.005 29.90 .339 -.004 30.85 .325 -.003 "'TITI'ITIMIMIETHfllflhljllllfl!I! RSITY Ill WI“ 42